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ELEMENTS 


DRAWING   AND   MENSURATION 


APPLIED   TO 


THE  MECHANIC  ARTS. 


BY   CHARLES   DAVIES,   LL.D., 

AUTHOR    or   FIRST   LESSONS    IN   ARITHMETIC,   ARITHMETIC,    ELEMENTARY   ALGEBRA, 

ELEMENTS  OF  SURVEYING,  ELEMENTS  OF  DESCRIPTIVE  GEOMETRY,  SHADES, 

SHADOWS,  AND  PERSPECTIVE,  ANALYTICAL  GEOMETRY, 

DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


NEW   YORK: 
PUBLISHED    BY   A.  S.  BARNES   &    CO., 

No.   .'51   JOHN    STREET. 

1846. 


:i:)3 


Entered  according  to  Act  of  Congress,  in  the  year  1846, 

By  CHARLES  DAVIES, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Southern 
District  of  New  York. 


StereotypeJ  by 

RICHARD  C.  VALENTINE, 

New  York. 


C.  A.  ALVORD,  Printer, 
Corner  of  John  and  Dutch  itreeti. 


PREFACE. 


The  design  of  the  present  work  is  to  afford  an  ele- 
mentary text-book  of  a  practical  character,  adapted  to  the 
wants  of  a  community,  where  every  day  new  demands 
arise  for  the  applications  of  science  to  the  useful  arts. 
There  is  little  to  be  done,  in  such  an  undertaking,  ex- 
cept to  collect,  arrange,  and  simplify,  and  to  adapt  the 
work,  in  all  its  parts,  to  the  precise  place  which  it  is 
intended  to  fill. 

The  introduction  into  our  schools,  within  the  last  few 
years,  of  the  subjects  of  Natural  Philosophy,  Astronomy, 
Mineralogy,  Chemistry,  and  Drawing,  has  given  rise  to 
a  higher  grade  of  elementary  studies ;  and  the  extended 
applications  of  the  mechanic  arts  call  for  additional  in- 
formation among  practical  men. 

To  understand  the  most  elementary  treatise  on  Natu- 
ral Philosophy,  or  the  simplest  work  on  the  Mechanic 
Arts,  or  even  to  make  a  plane  drawing,  some  knowledge 
of  the  principles  of  Geometry  is  indispensable  ;  and  yet, 
those  in  whose  hands  such  works  are  generally  placed, 
or  who  are  called  upon  to  make  plans  in  the  mechanic 
arts,  feel  that  they  have  hardly  time  to  go  through  with 
a  full  course  of  exact  demonstration. 

The  system  of  Geometry  is  a  connected  chain  of  rig- 
orous logic.  Every  attempt  to  compress  the  reasoning, 
by  abridging  it  at  the  expense  of  accuracy,  has  been  uni- 
formly and  strongly  condemned. 

111857 


IV  -  PREFACE. 

All  the  truths  of  Geometry  necessary  to  carry  out  fully 
the  plan  of  the  present  work,  are  made  accessible  to  the 
general  reader,  without  departing  from  the  exactness  of 
the  geometrical  methods.  This  has  been  done  by  omit- 
ting the  demonstrations  altogether,  and  relying  for  the 
impression  of  each  particular  truth  on  a  pointed  question 
and  an  illustration  by  a  diagram.  In  this  way,  it  is  be- 
lieved that  all  the  important  properties  of  the  geometrical 
figures  may  be  learned  in  a  few  weeks  ;  and  after  these 
properties  are  developed  in  their  practical  applications, 
the  mind  receives  a  conviction  of  their  truth  Httle  short 
of  what  is  afforded  by  rigorous  demonstration. 

The  work  is  divided  into  seven  Books,  and  each  book 
is  subdivided  into  sections. 

In  Book  I.  the  properties  of  the  geometrical  figures 
are  explained  by  questions  and  illustrations. 

In  Book  II.  are  explained  the  construction  and  uses 
of  the  various  scales,  and  also  the  construction  of  geo- 
metrical figures.  It  is,  as  its  title  imports.  Practical 
Geometry. 

Book  III.  treats  of  Drawing. — Section  I.,  of  the  Ele 
ments  of  the  Art ;   Section  II.,  of  Topographical  Draw- 
ing ;  and  Section  III.,  of  Plan-Drawing. 

Book  IV.  treats  of  Architecture, — explaining  the  dif- 
ferent orders,  both  by  descriptions  and  drawings. 

Book  V.  contains  the  application  of  the  principles  of 
Geometry  to  the  mensuration  of  surfaces  and  solids.  A 
separate  rule  is  given  for  each  case,  and  the  whole  is 
illustrated  by  numerous  and  appropriate  examples. 

Book  VI.  is  the  application  of  the  preceding  parts  to 
Artificers'  Work.  It  contains  full  explanations  of  all  the 
scales  and  measures  used  by  mechanics — the  construc- 
tion of  these  scales — the  uses  to  which  they  are  applied 


PREFACE.  V 

— and  specific  rules  for  the  calculations  and  computa- 
tions which  are  necessary  in  practical  operations. 

Book  VII.  is  an  introduction  to  Mechanics.  It  ex- 
plains the  nature  and  properties  of  matter,  the  laws  of 
motion  and  equilibrium,  and  the  principles  of  all  the  sim- 
ple machines. 

From  the  above  explanations,  it  will  be  seen  that  the 
work  is  entirely  practical  in  its  objects  and  character 
Many  of  the  examples  have  been  selected  from  a  small 
work  somewhat  similar  in  its  object,  recently  published 
in  Dublin,  by  the  Commissioners  of  National  Education, 
and  some  from  a  small  French  work  of  a  similar  charac- 
ter. Others  have  been  taken  from  Bonnycastle's  Men- 
suration, and  the  Library  of  Useful  Knowledge  was 
freely  consulted  in  the  preparation  of  Book  VII.  A 
friend,  Lt.  Richard  Smith,  also  furnished  most  of  the 
first  and  second  sections  of  Book  III.;  and  the  third 
section  was  chiefly  taken  from  an  "English  work. 

The  author  has  indulged  the  hope  that  the  present 
work,  together  with  his  First  Lessons  in  Arithmetic  for 
Beginners,  his  Arithmetic,  Elementary  Algebra,  and  Ele 
mentary  Geometry,  will  form  an  elementary  course  of 
mathematical  instruction  adapted  to  the  wants  of  Prac« 
tical  men,  Academies  and  the  higher  grade  of  schools. 

Wrst  Point,  Murcfi,  18-lfi. 


DAVIES' 
COURSE  OF  MATHEMATICS. 


DAVIES'  FIRST  LESSONS  IN  ARITHMETIC— For  beginnent 

DAVIES'  ARITHMETIC— Designed  for  the  use  of  Academies  and 
Schools. 

KEY  TO  DAVIES'  ARITHMETIC 

DAVIES'  UNIVERSITY  ARITHMETIC— Embracing  the  Science 
of  Numbers,  and  their  numerous  applications. 

KEY  TO  DAVIES'  UNIVERSITY  ARITHMETIC 

DAVIES'  ELEMENTARY  ALGEBRA— Being  an  Introduction  to 
the  Science,  and  forming  a  connecting  link  between  Arithmetic  and 
Algebra, 

KEY  TO  DAVIES'  ELEMENTARY  ALGEBRA. 

DAVIES'  ELEMENTARY  GEOMETRY.— This  work  embraces  the 
elementary  principles  of  Geometry.  Tiie  reasoning  is  plain  and  con- 
cise, but  at  the  same  time  strictly  rigorous. 

DxWIES'  ELEMENTS   OF  DRAWING  AND  MENSURATION 

— Applied  to  the  Mechanic  Arts. 

DAVIES'  BOURDON'S  v^LGEBRA— Including  Sturms'  Theorem,— 
Being  an  Abridgment  of  the  work  of  M.  Bourdon,  with  the  addition  of 
practical  examples. 

DAVIES'   LEGENDRE'S   GEOMETRY  and  TRIGONOMETRY. 

— Being  an  Abridgment  of  the  work  of  M.  Legendre,  with  the  addition 
of  a  Treatise  on  Mensuration  of  Planes  and  Solids,  and  a  Tabic  of 
Logarithms  and  LooARrniiMic  Sines. 

DAVIES'  SURVEYING— With  a  description  and  plates  of  the  Theod- 
olite, Compass,  Plane-Table,  and  Level  :  also.  Maps  of  the  Toro- 
GRAPHiCAL  Signs  adopted  by  the  Engineer  Department — an  explana- 
tion of  the  method  of  surveying  the  Public  Lands,  and  an  Elementary 
Treatise  on  Navigation. 

DAVIES'  ANALYTICAL  GEOMETRY— Embracing  the  Equa- 
tions OF  THE  Point  and  Straight  Line — of  the  Conic  Sections — of 
the  Line  and  Plane  in  Space — also,  the  discussion  of  the  General 
Equation  of  the  second  degree,  and  of  Surfaces  of  the  second  order. 

DAVIES'  DESCRIPTIVE  GEOMETRY,— With  its  application  to 
Spherical  Projections. 

DAVIES'  SHADOWS  and  LINEAR  PERSPECTIVE. 

PAVIES'  DIFFERENTLVL  and  INTEGRAL  CALCULUS. 


CONTENTS. 


BOOK   I.— SECTION  I. 

Page 

Of  Lines  and  Angles 13 

Of  Parallel  Lines — Oblique  Lines 14 

Of  Horizontal  Lines — Vertical  Lines 14 

Of  Angles  formed  by  Straight  Lines — By  Curves 15 

Of  the  Right  Angle — Acute  Angle — Obtuse  Angle 15—1 6 

Two  Lines  intersecting  each  other 16 — 17 

Parallels  cut  by  a  third  Line — Oblique  Lines 17 

Of  the  Circle,  and  Measurement  of  Angles 17 

Degrees  in  a  Right  Angle — Quadrant 18 

Sum  of  the  Angles  on  the  same  Side  of  a  Line 19 

Sum  of  the  Angles  about  a  Point 19 

SECTION  IL 

Plane  Figures 20 

Different  Kinds  of  Polygons 21 

Different  Kinds  of  Quadrilaterals 20—22 

Diagonal  of  a  Quadrilateral 23 

Square  on  the  Hypothenuse  of  a  Right-angled  Triangle  23 

SECTION  in. 

Of  the  Circle,  and  Lines  of  the  Circle 24 

Radius  of  the  Circle — Diameter  of  the  Circle 24 

Arc — Chord — Segment — Sector  25 

Angle  at  the  Centre — At  the  Circumference 26 

Angle  in  a  Segment — Secant — Tangent 26 

Figure  inscribed  in  a  Circle — Figure  circumscribed  about  it....  27 

Measure  of  an  Angle  at  the  Centre — At  the  Circumference  ...  27 — 26 

Sum  of  the  Angles  of  a  Triangle — Chords  of  the  Circle 28 — 29 


Vlll  CONTENTS. 

BOOK  II.— SECTION  I. 

•  Pago 

Practical  Geometry 30 

Description  of  Dividers,  and  Use 30 — 31 

Ruler  aaid  Triangle,  and  Use 32 — 33 

Scale  of  Equal  Parts,  and  Use 33 — ^34 

Diagonal  Scale  of  Equal  Fa  "s,  and  Use 36 — 37 

Scale  of  Chords,  and  Us*-    38 

Protractor,  and  Use 38 

Gunter'~  Scale 39 

Practical  Probleim   40—50 

Questions  for  P      Lice 50 — 52 

BOOK  III.— SECTION  I. 

Drawing  IN  i-Janeral 53 

Illustrat V  u  of  Form— Of  Shade  and  Shadow 53—60 

Manner  of  using  the  Pencil 60 — 61 

General  Remarks 61 — 63 

SECTION  II. 

Topographical  Drawing 63 

Description  of  Topographical  Drawing 63 

Explanation  of  the  Figures  and  Signs 64 — 70 

SECTION  III. 

Plan  Drawing 70 

Geometrical  Drawings — Defined 70 

Hcrisontal  Plane— Defined 70 

Vertical  Plane — Defined 71 

Plan — Denned... , 71 

Illustrations  of  Plan 71—78 

Sections  vvv 78 — 82 

The  Elevation ."..»:lL.fc. 82—86 

Remarks  on  Elevations 86 — 88 

Oblique  Elevations  88 — 95 

General  Remarks 95 — 96 

BOOK  IV.— SECTION  I. 

Of  Architecture  97 

Definition  of  Architecture — How  divided 97 

Elements  of  Architecture — Mouldings 97 — 100 


CONTENTS.  IX 

SECTION  II. 

Page 

Orders  of  Architecture — Their  Parts  , 102 — 104 

Tuscan  Order 105 

Doric  Order 105 

Ionic  Order 107 

Corinthian  Order 107 

BOOK  v.— SECTION  I. 

Mensuration  of  Surfaces  109 

Unit  of  Length,  or  Linear  Unit 109 

Unit  of  Surface,  or  Superficial  Unit 109 

Meaning  of  the  term  Rectangle 110 

Denominations  in  which  Areas  are  computed 112 — 114 

Area  of  the  Triangle 114—117 

Properties  of  the  Right-angled  Triangle 117 — 119 

Area  of  the  Square 119—120 

Area  of  the  Parallelogram 120 — 121 

Area  of  the  Trapezoid  121—122 

Area  of  the  Quadrilateral — Of  an  Irregular  Figure 122 — 125 

Areas  and  Properties  of  Polygons 125 — 132 

Of  the  Circle— Area  and  Properties 132—143 

Of  Circular  Rings 143_144 

Area  of  the  Ellipse 144—145 

SECTION  II. 

Mensuration  of  Solids 145 

Definition  of  a  Solid— Different  Kinds 145—147 

,    Content  of  Solids— Unit  of  Solidity— Table 147—149 

Of  the  Prism  149—152 

Of  the  Pyramid 152—157 

Of  the  Frustum  of  a  Pyramid  157—159 

-    Of  the  Cylinder 159—163 

Of  the  Cone ..V. 163—166 

Of  the  Frustum  of  a  Cone 167—169 

Of  the  Sphere 169—173 

Of  Spherical  Zones 174 

Of  Spherical  Segments 174—176 

Of  the  Spheroid 176—178 

Of  Cylindrical  Rings 178—179 

Of  tho  Five  Regular  Solids  179—183 

1* 


X  CONTENTS. 

BOOK  VI. 

Page 

Artificers'  Work 184 

SECTION  I. 

Of  Measures  184 

.    Carpenters'  Rule — Description  and  Uses 184 — 187 

To  multiply  Numbers  by  the  Carpenters'  Rule  187 — 190 

To  find  the  Content  of  a  Piece  of  Timber  by  the  Carpenters' 

Rule 190 

Table  for  Board  Measure 191 

Board  Measure  192 

SECTION  II. 

Of  Timber  Measure 193 

To  find  the  Area  of  a  Plank 193 194 

To  cut  a  given  Area  from  a  Plank 195 

To  find  the  solid  Content  of  a  square  Piece  of  Timber 195 — 197 

To  cut  off  a  given  Solidity  from  a  Piece  of  Timber  197 

To  find  the  Solidity  of  round  Timber 198 

Of  Logs  for  Sawing I99 

To  find  the  Number  of  Feet  of  Boards  which  can  be  sawed 

from  a  Log 200 

SECTION  IIL 

Bricklayers'  Work 201 

How  Artificers'  Work  is  computed 201 

Dimensions  of  Brick 202 

To  find  the  Number  of  Bricks  necesssuy  to  build  a  given  Wall  202 

Of  Cisterns 204 

To  find  the  Content  of  a  Cistern  in  Hogsheads 204 

Having  the  Height  of  a  Cistern,  to  find  its  Diameter  that  it 

may  contain  a  given  Quantity  of  Water 205 

Having  the  Diameter,  to  find  the  Height 205 

SECTION  IV. 

Masons' Work 20C 

SECTION  V. 

Carpenters'  and  Joiners'  Work , 207 

Of  Bins  for  Grain 2Qg 


CONTENTS.  XI 

Page 

To  find  the  Number  of  cubic  Feet  in  any  Number  of  Bushels  208 
To  find  the  Number  of  Bushels  which  a  Bin  of  a  given  Size 

wiUhold 208 

To  find  the  Dimensions  of  a  Bin  which  shall  contain  a  given 

Number  of  Bushels 209 

SECTION  VI. 

Slaters'  and  Tilers'  Work 210 

SECTION  VIL 

Plasterers'  Work 210 

To  find  the  Area  of  a  Cornice 211 

SECTION  VIIL 

Painters'  Work  212 

SECTION  IX. 

Pavers' Work 212 

SECTION  X 

Plumbers'  Work  213 

BOOK  VII. 

Introduction  to  Mechanics 215 

SECTION  I. 

Op  Matter  and  Bodies 215 

Matter— Defined 215 

Body— Defined  215 

Space — Defined 215 

Of  the  Properties  of  Bodies 215 

Impenetrability — Defined 215 

Extension — Defined  :. 216 

Figure— Defined 216 

Divisibility— Defined 216 

Inertia — Defined 217 

Atoms — Defined 217 

Attraction  of  Cohesion 217 

Attraction  of  Gravitation  218 

Weight— Defined  819 


Xll  CONTENTS. 

SECTION  II. 

Page 

Laws  of  Motion,  and  Centre  of  Gravity &19 

Motion— Defined 219 

Force  or  Power — Defined 219 

Velocity— Defined 219 

Momentum — Defined 221 

Action  and  Reaction — Defined.. 221 

Centre  of  Gravity— Defined 222 

SECTION  III. 

Of  the  Mechanical  Powers 224 

General  Principles 224 

Lever — Different  Kinds  224 — 227 

Pulley 227—229 

Wheel  and  Axle 230 

Inclined  Plane 231 

Wedge — Screw 232 

General  Remarks  '. 233 

SECTION  IV. 

Of  Specific  Gravity 234 

Specific  Gravity — Defined '. 234 

When  a  Body  is  specifically  heavier  or  lighter  than  another  234 

Density— Defined 234 

To  find  the  Specific  Gravity  of  a  Body  heavier  than  Water  236 

To  find  the  Specific  Gravity  of  a  Body  lighter  than  Water  -  237 

To  find  the  Specific  Gravity  of  Fluids  ....'. 238 

Table  of  Specific  Gravities 239 

To  find  the  Solidity  of  a  Body  when  its  Weight  and  Specific 

Gravity  are  known 240 


GEOMETRY 


BOOK  I. 

SECTION  I. 

OF    LINES    AND    ANGLES. 

1.  What  is  a  line? 

A  Line  is  length,  without  breadth  or  thickness. 

2.  What  are  the  extremities  of  a  line  called? 

The  Extremities  of  a  Line  are   called  points ;    and  any 
place  between  the  extremities,  is  also  called  a  point. 

3.  What  is  a  straight  line  ? 

A  StraigJit  Line,  is  the  shortest  dis- 
tance from  one  point  to  another.  Thus, 
AB  is  a  straight  line,  and  the  shortest 
distance  from  A  to  B. 

4.  What  is  a  curve  lijie  ? 

A  Curve  Line,  is  one  which  changes 
Its  direction  at  every  point.  Thus, 
ABC  is  a  curve  line. 

5.  What  does  the  word  line  mean  ? 

The  word  Line,  when  used  by  itself,  means  a  straight 
line ;  and  the  word  Curve,  means  a  curve  line. 


14  BOOK    I. SECTION    I. 

6.  What  is  a  surface  ? 

A  Surface  is  that  which  has  length  and  breadth,  without 
height  or  thickness. 

7.  What  is  a  plane,  or  plane  surface  ? 

A  Plane  is  that  which  lies  even  throughout  its  whole  ex- 
tent, and  with  which  a  straight  line,  laid  in  any  direction, 
will  exactly  coincide. 

8.  When  are  lines  said  to   be  parallel? 
Two  straight  lines  are  said  to  be  paral-. 

lei  when  they  are   at  the   same  distance 

from  each  other  at  every  point.  Parallel 
lines  will  never  meet  each  other. 

9.  When  are  two  curves  said  to  be  parallel? 
Two  curves  are  said  to  be  parallel  or 

concentric,  when  they  are  at  the  same  dis- 
tance from  each  other.  Parallel  curves 
will  not  meet  each  other. 

10.  What  are  oblique  lines? 

Oblique  lines  are  those  which  ap- 
proach each  other,  and  meet  if  suffi- 
ciently prolonged. 

11.  What  are  horizontal  lines? 

Lines  which  are  parallel  to  the  horizon,  or  to  the  water 
level,  are  called  Horizontal  Lines.  Thus,  the  eaves  of  a 
house  are  horizontal. 

12.  What  are  vertical  lines? 

All  plumb  lines  are  called  Vertical  Lines.  Thus,  trees 
and  plants  grow  in  vertical  lines. 

13.  What  is  an  angle?     How  is  it  read? 

An  Angle  is  the  opening  or  inclination  of  two  lines  which 


OF    LINES    AND    ANGLES. 


15 


meet  each  other  in  a  point.  Thus  the 
lines  A  C,  AB,  form  an  angle  at  the  point 
A.  The  lines  AC,  and  AB,  are  called 
the  sides  of  the  angle,  and  their  intersec- 
tion A,  the  vertex. 

The  angle  is  generally  read  by  placing  the  letter  at  the 
vertex  in  the  middle :  thus,  we  say  the  angle  CAB.  We 
may,  however,  say  simply,  the  angle  A. 


14.  May    angles    be  formed   by  curved 
lines  ? 

Yes,    either   by  two    curves,    CA,  BA    -^^ 
forming  the  angle  A,  called  a  curvilinear 
angle : 


Or,  by  two  curves  AC,  AB,  forming      . 
the  angle  A: 


Or,  by  a  straight  line  and  curve,  which 
is  called  a  mixtilinear  angle. 


D 


15.  When  is  one  line  said  to  be  perpendicular  to  another  ? 

One  line  is  perpendicular  to  another, 
when   it  inclines  no  more   to  the  one 
side  than  to  the  other.     Thus,  the  line 
DB  is  perpendicular  to  AC,  and   the    j[ 
angle  DBA  is  equal  to  DBC. 


16 


BOOK    I. SECTION    I. 


16.    What  are  the  angles  called? 

When  two  lines  are  perpendicular  to 
each  other,  the  angles  which  they  form 
are  called  right  angles.  Thus,  DBA 
and  DBC  are  right  angles.  Hence,  all 
right  angles  are  equal  to  each  other. 


D 


B 


17.  What  is  an  acute  angle? 

An  acute  angle  is  less  than  a  right 
angle.     Thus,  DBC  is  an  acute  angle. 

1 8.  What  is  an  obtuse  angle  ? 

An  obtuse   angle   is   greater  than   a 


right  angle, 
angle. 


Thus,  EBC  is  an  obtuse 


19,  If  two  lilies  intersect  each  other,  what  follows 


If  two  lines  intersect  each  other, 
the  opposite  angles  A  and  A  are 
called  vertical  angles.  These  an- 
gles are  equal  to  each  other,  and 
so  also  are  the  opposite  angles  B 
and  B. 


M^ 


yi 


D 

"C 

E 


A 
C 


-B 
-D 


20.  What  follows  when  two  parallel  lines  are  cut  hy  a  third 
line  ? 

If  two  parallel  lines  CD,  AB,  are 
cut  by  a  third  line  IG,  the  angles 
IHD  and  AFG,  are  called  alternate 
angles.  These  angles  arc  equal  to 
each  other.     The  angle  IHD  is  also  G 

equal  to  the  angle  IFB,  and  to  the  opposite  angle  CHG. 

21.  What  follows  when  a  line  is  perpendicular  to  one  of 
several  parallel  lines  ? 

If  a   line   be   perpendicular  to  one   of  several  parallel 


OF    LINES   AND    ANGLES. 


17 


H 


lines,  it  will  be   perpendicular  to   all 
the    others.      Thus,  if  AB,    CD,  and     ^. 
EF,  be  parallel,  the   line   CH  drawn 
perpendicular  to  AB,  will  also  be  per-     ^ 
pendicular  to   CD  and  EF.  E 

22.   How  many  lines  can  he  drawn  from  one  point  perpen- 
dicular to  a  line  ? 

From  the  same  point  D,  only  one 
line  DB,  can  be  drawn,  which  will 
be  perpendicular  to  AB.  ^ 


D 


B    C 


23.   If  oblique  lines  are  also  drawn,  what  follows  ? 

If  oblique  lines  be  drawn,  as  DC,  DF,  then: — 

1st.  The  perpendicular  DB,  will  be  shorter  than  any  of 
the  oblique  lines. 

2d.  The  oblique  lines  which  are  nearest  the  perpendic- 
ular, will  be  less  than  those  which  are  more  remote. 


OF    THE    CIRCLE    AND    MEASUREMENT    OF    ANGLES. 


24.  What  is  the  circumference  of  a  circle 
The  circumference  of  a  circle  is  a 
curve  line,  all  the  points  of  which  are 
equally  distant  from  a  certain  point 
within,  called  the  centre.  Thus,  if  all 
the  points  of  the  curve  AEB  are  equal- 
ly distant  from  the  centre  C,  this  curve 
will  be  the  circumference  of  a  circle. 


Tr^M^ 


25.  For  what  is  the  circumference  of  a  circle  used  ? 
The  circumference  of  a  circle  is  used  for  the  measure- 
ment of  angles. 

26.  How  is  it  divided  ? 

It  is  divided  into  360  equal  parts  called  degrees,  each 


18 


BOOK    I. SECTION    I. 


degree  is  divided  into  60  equal  parts  called  minutes,  and 
each  minute  into  60  equal  parts  called  seconds.  The  de- 
grees, minutes,  and  seconds,  are  marked  thus,  °,  '',  ^' ;  and 
9°  18^  10^^,  are  read,  9  degrees,  18  minutes,  and  10  sec- 
onds. 

27.  How  are  the  angles  measured? 

Suppose  the  circumference  of  a  cir- 
cle to  be  divided  into  360  equal  parts, 
beginning  at  the  point  B.  If,  through 
the  point  of  division  marked  40,  we 
draw  CE,  then,  the  angle  ECB  will 
be  equal  to  40  degrees.  If  we  draw 
CF  through  the  point  of  division  marked  80,  it  will  make 
CB  an  angle  equal  to  80  degrees. 

28.  How  many  degrees   are   there  in  one  right  angle^ — in 
two — in  three — in  four  ? 

If  two  lines  AB,  DE,  are  perpen- 
dicular to  each  other,  the  four  angles 
BCD,  DCA,  ACE,  and  ECB,  will  be 
equal.  These  two  lines  will  divide  the 
circumference  of  the  circle  into  the 
four  equal  parts  BD,  DA,  AE,  and 
EB,  and  each  part  will  measure  one 
of  the  right  angles.  But  one  quarter  of  360  degrees,  is  90 
degrees.  Hence,  one  right  angle  contains  90  degrees,  two 
right  angles  180  degrees,  three  right  angles  270  degrees, 
and  four  right  angles  360  degrees. 

29.  What   is   07ie   quarter   of  the   circumference   called? — 
one  half  of  it  ? 

One  quarter  of  the  circumference  is 
called  a  quadrant,  and  contains  90  de- 
grees. One  half  of  the  circumference 
is  called  a  semi-circumference,  and  con- 


OF    LINES    AND    ANGLES.  19 

tains  180  degrees.     Thus,  AC  is  a  quadrant,  and  ACB  is 
a  semi-circumference. 

30.  If  one  straight  line  meets  another ^  what  is  the  sum 
of  the  two  angles  equal  to  ? 

If  a  straight  line  EB  meets  anoth- 
er  straight  line  AC,  the   sum  of  the 

angles  ABE  aiid  EBC,  will  be  equal     -J N^ 

to  two  right  angles,  since  these  two 

angles  are  measured  by  half  the  circumference. 

31.  If  there   be   several   angles,   what   will    their   sum   he 
equal  to? 

If  there   be    several   angles    CBF,  e  ^i 

FBE,  EBD,  DBA,  formed  on  the 
same  side  of  a  line,  their  sum,  for  a 
like  reason,  will  be  equal  to  two  right      ^  HA 

angles. 

32.  What   is   the  sum  of  all   the  angles  formed  about   a 
point  equal  to  ? 

The  sum  of  all  the  angles  ACB,  BCD, 
DCA,  which  can  be  formed  about  any 
point  as  C,  is  equal  to  four  right  angles, 
or  360  degrees,  since  they  are  measured 
by  the  entire  circumference. 


20 


BOOK    I. SECTION    II, 


SECTION   II. 


OF    PLANE    FIGURES. 


1.  WJmt  is  a  "plane  figure! 

A  plane  figure  is  a  portion  of  a  plane,  terminated  on  all 
sides  by  lines,  either  straight  or  curved. 

2.  When  the  hounding  lines  are  straight,  what  is  it  called? 
If  the  bounding  lines  are  straight,  the  space  they  enclose 

is  called  a  rectilineal  figure,  or  polygon. 

3.  What  are  the  lines  themselves  called  ? 

The  lines  themselves,  taken  together,  are  called  the  pe- 
rimeter of  the  polygon.  Hence,  the  perimeter  of  a  polygon 
is  the  sum  of  all  its  sides. 

4.  Nam^  the  different  kinds  of  polygons. 


A  polygon  of  three  sides,  is  called  a 
triangle. 


A  polygon  of  four  sides,  is   called   a 
quadrilateral. 


A  polygon  of  five   sides,  is  called   a 
pentagon.  \ 


OF    PLANE    FIGURES. 


21 


A  polygon  of  six   sides,  is  called   a 
hexagon. 


A  polygon  of  seven  sides,  is  called  a  heptagon. 
A  polygon  of  eight  sides,  is  called  an  octagon. 
A  polygon  of  nine  sides,  is  called  a  nonagon. 
A  polygon  of  ten  sides,  is  called  a  decagon. 
A  polygon  of  twelve  sides,  is  called  a  dodecagon. 

5.  What  is  the  perimeter  of  a  polygon  ? 

The  perimeter  of  a  polygon  is  the  sum  of  all  its  sides. 

6.  What  is   the   least  number  of  straight  lines  which  can 
enclose  a  space  ? 

Three  straight  lines,  are  the  smallest  number  which  can 
enclose  a  space. 

7.  Name  the  several  kinds  of  triangles. 

First.- — An  equilateral  triangle,  which 
has  its  three  sides  all  equal. 


Second. — An  isosceles  triangle,  which 
has  two  of  its  sides  equal. 


Third.— A  scalene  triangle,  which  has 
its  three  sides  all  unequal. 


-V/ 


22  BOOK    I. SECTION    II. 

Fourth. — A  right-angled  triangle,  which 
has  one  right  angle.  In  the  right-angled 
triangle  BAC,  the  side  BC  opposite  the 
right  angle,  is  called  the  hypothenuse.         j^ 

8.  What  is  the  base  of  a  triangle  ? — what  its  altitude  ? 
The  base  of  a  triangle   is   the   side  on  which'  it  stands. 

Thus,  BA  is  the  base  of  the  right-angled  triangle  BAC. 
The  line  drawn  from  the  opposite  angle  perpendicular  to 
the  base,  is  called  the  altitude.     Thus,  AC  is  the  altitude. 

9.  Name  the  different  kinds  of  quadrilaterals. 


First. — The  square,  which  has  all  its 
sides  equal,  and  all  its  angles  right  an- 
gles. 

Second. — The  rectangle,  which  has  its 
angles  right  angles,  and  its  opposite  sides 
equal  and  parallel. 

Third. — The  parallelogram,  which  has 
its  opposite  sides  equal  and  parallel,  but 
its  angles  not  right  angles. 

Fourth. — The  rhombus,  which  has  all 
its  sides  equal,  and  the  opposite  sides 
parallel,  without  having  its  angles  right 
angles. 

Fifth, — The  trapezoid,  which  has  only 
twi)  of  its  sides  parallel. 


10.   What  is  the  base  of  a^ figure?      What  its  altitude? 
The  base  of  a  figure  is  the  side  on  which  it  stands,  and 


OP    PLANE    FIGURES. 


23 


the  altitude  is  a  line  drawn  from  the  top,  perpendicular  to 
the  base. 


11.    What  is  a  diagonal? 

A  diagonal,  is  a  line  joining  the 
vertices  of  two  angles  not  adjacent. 
Thus,  AB  is  a  diagonal. 


12.  What  is  the  square  described  on  the  hypothenuse  of  a 
right-angled  triangle  equal  to? 

In  every  right-angled  triangle,  the   square  described  on 
the  hypothenuse,  is  equal  to 
the   sum  of  the   squares   de- 
scribed on  the  other  two  sides. 

Thus,  if  ABC  hQ  ^  right- 
angled  triangle,  right-angled 
at  C,  then  will  the  square  D, 
described  on  AB,  be  equal  to 
the  sum  of  the  squares  E 
and  F,  described  on  the  sides 
CB  and  AC.  This  is  called 
the  carpenter's  theorem. 

By  counting  the  small  squares  in  the  large  square  D, 
you  will  find  their  number  equal  to  that  contained  in  the 
small  squares  F  and  E.  ^^ 


D 

\i 


^i>^ 


24 


BOOK    I. SECTION    III. 


SECTION  III. 


OF    THE    CIRCLE,    AND    LINES    OF    THE    CIRCLE. 


1.  What  is  a  circle?  What  is  a  circumference  1 
A  circle  is  a  plane  figure,  bounded 
by  a  curve  line,  all  the  points  of  which 
are  equally  distant  from  a  certain  point 
within,  called  the  centre.  The  curve  line 
is  called  the  circumference.  Thus,  the 
space  enclosed  by  the  curve  ABD  is 
called  a  circle  :  the  curve  ABD  is  the 
circumference^  and  the  point  C,  the  centre. 


2.  What  is  the  radius  of  a  circle  ?     Are  all  radii  equal  <* 
Any  line,  as  CA,  drawn  from  the  cen-  . 

tre  C  to  the  circumference,  is  called  a 

radius i  and  two  or  more  such  lines,  are 

radii. 

All  the  radii  of  a  circle  are  equal  to 

each  other. 


3.   What  is  the  diameter  of  a  circle 
the  circumference? 

The  diameter  of  a  circle  is  any 
line,  as  AD,  passing  through  the 
centre  and  terminating  in  the  cir- 
cumference. Every  diameter  of  a 
circle  divides  it  into  two  equal 
parts,  called  semicircles,  or  half  cir- 
cles. 


How  does  it  divide 


OF    THE    CIRCLE,    ETC. 


35 


4.   What  is  an  arc? 
An  arc  is  any  portion  of  the  circum- 
ference.    Thus,  AEB  is  an  arc. 


5.   What  is  a  chord? 

A  chord  of  a  circle,  is  a  line  drawn 
within  a  circle,  and  terminating  in  the 
circumference,  but  not  passing  through 
the  centre.     Thus,  AB  is  a  chord. 

A  chord  divides  the  circle  into  two 
unequal  parts. 


6.    What  is  a  segment? 

A  segment  of  a  circle,  is  a  part  cut 
off  by  a  chord.  Thus,  AEB  is  a  seg- 
ment. 

The  part  AOB,  is  also  a  segment, 
although  the  term  is  generally  applied 
to  the  pai:t  which  is  less  than  a  semi- 
circle. 


7.   What  is  a  sector? 

A  sector  of  a  circle,  is  any  part  of  a 

circle  bounded  by  two  radii  and  the  arc 

included  between  them.     Thus,  ACB  is 

a  sector. 

2 


26 


BOOK    I. SECTION    III. 


8.   What  is  an  angle  at  the  centre? 

An  angle  at  the  centre,  is  one  whose 
vertex  is  at  the  centre  of  the  circle. 
Thus,  BCE,  or  ECD,  is  an  angle  at 
the  centre. 


9.  What  is  an  angle  at  the  circumfe- 
rence 1 

An  angle  at  the  circumference,  is  one 
whose  angular  point  is  in  the  circum- 
ference. Thus,  BAG,  or  BOC,  is  an 
angle  at  the  circumference. 


seg- 


10.     What    is    an    angle    ir 
ment  ? 

An  angle  in  a  segment,  is  formed  by 
two  lines  drawn  from  any  point  of  the 
segment  to  the  two  extremities  of  the 
arc.  Thus,  ABE  is  an  angle  in  a  seg- 
ment. 


11.   What  is  a  secant  line?  A 

A  secant  line,  is  one  which  meets  the 
circumference  in  two  points,  and  lies 
partly  within  and  partly  without.  Thus, 
AB  is  a  secant  line. 


12,    What  is  a  tangent  line? — What  position  has  it  with 
the  radius  passing  through  the  point  of  contact  ? 

A  tangent  is  a  line  which  has  but  one  point  in  com- 


OF    THE    CIRCLE,    ETC. 


1^ 


mon  with  the  circumference.  Thus, 
EMD  is  a  tangent.  The  point  M 
at  which  the  tangent  touches  the 
circumference  is  called  the  point  of 
contact.  The  tangent  line  is  perpen- 
dicular to  the  radius  passing  through 
the  point  of  contact.  Thus,  CM  is 
perpendicular  to  EMD. 


13.   When  is  a  figure  said  to  be  inscribed  in  a  circle  ?- 
What  is  said  of  the  circle  ? 

A  figure  is  said  to  be  inscribed  in 
a  circle  when  all  the  angular  points 
of  the  figure  are  in  the  circumference. 
The  circle  is  then  said  to  circumscribe 
the  figure.  Thus,  the  triangle  ABC  is 
inscribed  in  the  circle,  and  the  circle 
circumscribes  the  triangle. 


14.    When  is  a  figure  said   to   be  circumscribed  about  a 
circle  1 

A  figure  is  said  to  be  circumscribed 
about  a  circle,  when  all  the  sides  of  the 
figure  touch  the  circumference.  The  cir- 
cle is  then  said  to  be  inscribed  in  the 
figure. 


15.  How  is  an  angle  at  the  centre  of  a  circle  measured? 

An  angle  at  the  centre  of  a  circle  is 
measured  by  the  arc  contained  by  the 
sides  of  the  angle.  This  arc  is  said  to 
subtend  the  angle.  Thus,  the  angle  ^C5 
is  measured  by  the  degrees  in  the  arc 
ABB,  and  is  subtended  by  the  arc  AEB. 


28 


BOOK    I. SECTION    III. 


16.   What  measures  an  angle  at  the  circumference  1 

kx\.  angle  at  the  circumference  of  a  circle,  is  measured 

by   half   the    arc    which   subtends    it. 

Thus,  the  angle  BAT)  is  measured  by 

half  the  arc   BT).     Hence,  it  follows, 

that  when  an  angle  at  the  centre  and 

an  angle   at  the   circumference   stand 

on  the  same  arc  jBD,  the  angle  at  the 

centre  will  be  double  the  angle  at  the 

circumference. 


1 7.  What  is  the  sum  of  the  three  an- 
gles of  any  triangle  equal  to  ? 

To  180  degrees,  since  they  will  be 
measured  by  one  half  of  the  entire  cir- 
cumference. 


18.    What  is  an  angle  in  a  semicircle  equal  to . 


An  angle  inscribed  in  a  semicircle. 
Thus,  if  AB  be  the  diameter  of  a  cir- 
cle, then  will  the  angle  ACB  be  equal 
to  90  degrees.  This  angle  is  measured 
by  one  half  the  semi-circumference,  that 
is,  by  one  half  of  180°,  or  by  90°. 


is   a   right  angle. 


19.  Are   the   arcs  intercepted   hy  parallel  chords  equal,  or 
unequal  ? 

Two  parallel  chords  intercept  equal 
arcs.     That  is,  if  the  chords  AB  and 

CD  are  parallel,  the   arcs  AC   and    c-( yjf 

DB,   which  they   intercept,   will  be 
equal. 


OF    THE    CIRCLE,    ETC. 


29 


20.  If  a  line  be  drawn  from  the  centre  of  a  circle  perpen- 
dicular to  a  chord,  what  follows  1 

If  from  the  centre  of  a  circle 
a  line  be  drawn  perpendicular  to 
a  chord,  it  will  bisect  the  chord, 
and  also  the  arc  of  the  chord. 
Thus,  CFE  drawn  from  the  cen- 
tre C,  perpendicular  to  AB,  bi- 
sects AB  at  F,  and  also  makes 
AE  =  EB. 


21.  How  is  the  distance  from  the  centre  of  a  circle  to  a 
chord  measured? 

The  distance  from  the  centre  of  a  circle  to  a  chord,  is 
measured  on  a  perpendicular  to  the  chord. 


22.   How  are   chords  which   are   equally  distant  from  the 
centre  ? 

In  the  same,  or  in  equal  circles, 
chords  which  are  equally  distant  from 
the  centre,  are  equal.  Thus,  if  CA 
=  CB,  then  will  the  chord  FG  = 
chord  DE. 


h.'. 


30 


BOOK    II. SECTION    I. 


BOOK  II. 


SECTION   I. 


PRACTICAL    GEOMETRY. 


1.  What  is  Practical  Geometry? 

Practical  geometry  explains  the  methods  of  constructing, 
or  describing  the  geometrical  figures. 

2.  What  is  a  problem  1 

Any  question  which  requires  something,  to  be  done ;  and 
doing  the  thing  required,  is  called  the  solution  of  the  prob- 
lem. 

3.  What  are  necessary  in  the  solution  of  geometrical  prob- 
lems ? 

Certain  instruments  which  are  now  to  be  described. 

4.  What  are  the  dividers  or  compasses? 


The  dividers  is  the   most  simple   and  useful  of  the   in- 


PROBLEMS.  31 

stniments  used  for  describing  figures.  It  consists  of  two 
legs,  ha  and  &e,  which  may  be  easily  turned  around  a 
joint  at  b. 

5.  How  will  you  lay  off  on  a  line,  as  CD,  a  distance  equal 
to  AB? 

Take  up  the  dividers  with  the  thumb  and  second  finger, 
and  place  the  fore-finger  on  the  joint  at  h.     Then,  set  one 
foot  of  the  dividers  at  A,  and  ex- 
tend the  other  leg  with  the  thumb       ^ 5 

and  fingers,  until  the  foot  reaches       c  U       jy 

to  B.     Then,  raising  the  dividers, 

place  one  foot  at  C,  and  mark  with  the  other  the  distance 
CE,  this  will  evidently  be  equal  to  AB. 

6.  How  will  you  describe  from  a  given  centre,  the  circum- 
ference of  a  circle  having  a  given  radius  ? 

Let  C  be  the  given  centre,  and  CB  the  given  radius. 
Place  one  foot  of  the  dividers  at  C,  and 
extend  the  other  leg  until  it  shall  reach 
to  B.  Then  turn  the  dividers  around  the 
leg  at  C,  and  the  other  leg  will  describe 
the  required  circumference. 

7.  How  may  this  be  done  on  a  black  board  with  a  string 
and  chalk? 

Take  one  end  of  the  string  between  the  thumb  and  fore- 
finger of  the  left  hand,  and  place  it  at  the  centre  C.  Then 
take  the  length  of  the  radius  on  the  string,  at  which  point 
place  the  chalk  held  between  the  thumb  and  finger  of  the 
right  hand.  Then,  holding  the  end  of  the  string  firmly  at 
C,  turn  the  right  hand  around,  and  the  chalk  will  trace 
the  circumference  of  the  circle. 


32  BOOK    II. SECTION    I. 

8.  Describe  the  ruler  and  triangle. 


A  ruler  of  a  convenient  size,  is  about  twenty  inches  in 
length,  two  inches  wide,  and  one-fifth  of  an  inch  in  thick- 
ness. It  should  be  made  of  a  hard  material,  perfectly- 
straight  and  smooth. 

The  hypothenuse  of  the  right-angled  triangle,  which  is 
used  in  connection  with  it,  should  be  about  ten  inches  in 
length,  and  it  is  most  convenient  to  have  one  of  the  sides 
considerably  longer  than  the  other.  We  can  resolve  with 
the  ruler  and  triangle  the  two  following  problems. 

9.  Describe  the  manner  of  draiving  through  a  given  point 
a  line,  which  shall  be  parallel  to  a  given  line^  with  the  ruler 
and  triangle. 

Let  C  be  the  given  point,  and  AB  the  given  line. 

Place  the  hypothenuse  of  the  tri-  q 

angle  against  the  edge  of  the  ruler,       ' 

and  then  place  the  ruler  and  trian-     ^/^ ^ 

gle  on  the  paper,  so  that  one  of  the 

sides  of  the  triangle  shall  coincide  exactly  with  AB — the 

triangle  being  below  the  line  AB. 

Then  placing  the  thumb  and  lingers  of  the  left  hand 
firmly  on  the  ruler,  slide  the  triangle  with  the  other  hand 
along  the  ruler  until  the  side  which  coincided  wdth  AB 
reaches  the  point  C.     Leaving  the  thumb  of  the  left  hand 


SCALE    OF    EQUAL    PARTS.  33 

on  the  ruler,  extend  the  fingers  upon  the  triangle  and  hold 
it  firmly,  and  with  the  right  hand  mark  with  a  pen  or  pen- 
cil a  line  through  C :  this  line  will  be  parallel  to  AB. 

10.  Explain  the  manner  of  drawing  through  a  given  point, 
a  line  which  shall  be  perpendicular  to  a  given  line,  with  the 
ruler  and  triangle. 

Let  AB  be  the  given  line,  and  D  the  given  point. 

Place,  as  before,  the  hypothenuse 
of  the  triangle  against  the  edge  of  the                  .    . 
ruler.     Then  place  the  ruler  and  tri- 
angle so  that  one  of  the  sides  of  the      4 1 ^ 

triangle  shall  coincide  exactly  with 

the  line  AB.  Then  slide  the  triangle  along  the  ruler  until 
the  other  side  reaches  the  point  D.  Draw  through  D  a 
straight  line,  and  it  will  be  perpendicular  to  AB. 

11.  What  is  a  scale  of  equal  parts? 


A  scale  of  equal  parts  is  formed  by  dividing  a  line  of  a 
given  length,  into  equal  portions. 

If,  for  example,  the  line  ab,  of  a  given  length,  say  one 
inch,  be  divided  into  any  number  of  equal  parts,  as  10,  the 
scale  thus  formed  is  called  a  scale  often  parts  to  the  inch. 

12.    What  is  the  unit  of  a  scale,  and  how  is  it  laid  off? 

The  line  ab,  which  is  divided,  is  called  the  unit  of  the 
scale.  This  unit  is  laid  off  several  times  on  the  left  of  the 
divided  line,  and  the  points  marked  1,2,  3,  &c.  The  unit 
of  scales  of  equal  parts,  is,  in  general,  either  an  inch  or 
an  exact  part  of  an  inch.  If,  for  example,  the  unit  of  the 
scale  ab,  were  one  inch,  the  scale  would  be  one  of  ten 
parts  to  the  inch ;  if  it  were  half  an  inch,  the  scale  would 

2* 


34  BOOK    II. — SECTION    I. 

be  one  of  ten  parts  to  half  an  inch,  or  of  20  parts  to  the 
inch. 

13.  How  will  you  take  from  the  scale  two  inches  and  six- 
tenths  1 

Place  one  foot  of  the  dividers  at  2  on  the  left,  and  ex- 
tend the  other  to  .6,  which  marks  the  sixth  of  the  small 
divisions :  the  dividers  will  then  embrace  the  required  dis- 
tance. 

14.  How  will  you  lay  down,  on  paper,  a  line  of  a  given 
length,  so  that  any  number  of  its  parts  shall  correspond  to  the 
unit  of  the  scale  ? 

Suppose  that  the  given  line  were  75  feet  in  length,  and 
it  were  required  to  draw  it  on  paper,  on  a  scale  of  25  feet 
to  the  inch. 

The  length  of  the  line,  75  feet,  being  divided  by  25,  will 
give  3,  the  number  of  inches  which  will  represent  the  line 
on  paper. 

Therefore,  draw  the  indefinite  line  AB^  on  which  lay  off 


ji 


a  distance  AC  equal  to  3  inches:  AC  will  then  represent 
the  given  line  of  75  feet,  drawn  to  the  required  scale. 

15.    What  does  the  last  question  explain? 

The  last  question  explains  the  method  of  laying  down  a 
line  upon  paper,  in  such  a  manner  that  a  given  number  of 
parts  shall  correspond  to  the  unit  of  the  scale,  whether  that 
unit  be  an  inch  or  any  part  of  an  inch. 

When  the  length  of  the  line  to  be  laid  down  is  given, 
and  it  has  been  determined  how  many  parts  of  it  are  to 
be  represented  on  the  paper  by  a  distance  equal  to  the 
unit  of  the  scale,  we  find  the  length  which  is  to  be  taken 
from  the  scale  by  the  following 


PROBLEMS.  -65 


RULE. 


Divide  the  length  of  the  line  by  the  number  of  parts  which  is 
to  be  represented  by  the  unit  of  the  scale :  the  quotient  will 
show  the  number  of  parts  which  is  to  be  taken  from  the 
scale. 

EXAMPLES. 

1.  If  a  line  of  640  feet  in  length  is  to  be  laid  down  on 
paper,  on  a  scale  of  40  feet  to  the  inch ;  what  length  must 
be  taken  from  the  scale  ? 

40)640(16  inches. 

2.  If  a  line  of  357  feet  is  to  be  laid  down  on  a  scale 
of  68  feet  to  the  unit  of  the  scale,  (which  we  will  suppose 
half  an  inch,)  how  many  parts  are  to  be  taken? 

(  5.25,  parts,  or 
^^^'    i  2.625  inches. 

16.  When  the  length  of  a  line  is  given  on  the  paper,  how 
will  you  find  the  true  length  of  the  line  represented  ? 

Take  the  line  in  the  dividers  and  apply-  it  to  the  scale, 
and  note  the  number  of  units,  and  parts  of  a  unit  to  which 
it  is  equal.  Then,  multiply  this  number  by  the  number  of 
parts  which  the  unit  of  the  scale  represents,  and  the  pro- 
duct will  be  the  length  of  the  line. 

EXAMPLES. 

1.  Suppose  the  length  of  a  line  drawn  on  the  paper,  to 
be  3.55  inches,  the  scale  being  40  feet  to  the  inch :  then, 

3.55  X  40  =  142  feet,  the  length  of  the  line. 

2.  If  the  length  of  a  line  on  the  paper  is  6.25  inches, 
and  the  scale  be  one  of  30  feet  to  the  inch,  what  is  the 
true  length  of  the  line  ?  ''"^     *" 

Ans.  187.5  feet. 


36 


BOOK    II. — SECTION    I. 


17.  How  do  you  construct  the  diagonal  scale  of  equal 

PARTS  ? 


% 


df 


. 

w          \  w  \  \ 

o;> 

\    >                 M    1    /    ! 

08 

1  i   1    ,    ,  y(  M   1 

07 

1  1   1   I   i   I   U   I 

06 

n  1  1  n    I 

05 

i  hi  n       , 

04 

MM'     / 

03 

i  1  11     1     i 

02 

!  '       1 — 11 

01 

ii     ,-r-fii- 

a  .1  .2.3.4  .5  .G.l.S  .9 


This  scale  is  thus  constructed.  Take  ab  for  the  unit 
of  the  scale,  which  may  be  one  inch,  i,  ^,  or  |  of  an 
inch,  in  length.  On  ab  describe  the  square  abed.  Divide 
the  sides  ab  and  dc  each  into  ten  equal  parts.  Draw  af 
and  the  other  nine  parallels  as  in  the  figure. 

Produce  ab  to  the  left,  and  lay  off  the  unit  of  the  scale 
any  convenient  number  of  times,  and  mark  the  points  1, 
2,  3,  &c.  Then,  divide  the  line  ad  into  ten  equal  parts, 
and  through  the  points  of  division  draw  parallels  to  ab  as 
in  the  figure. 

Now,  the  small  divisions  of  the  line  ab  are  each  one- 
tenth  (.1)  of  ab ;  they  are  therefore  .1  of  ad,  or  .1  of  ag 
or  gh. 

If  we  consider  the  triangle  adf,  the  base  df  is  one-tenth 
of  ad,  the  unit  of  the  scale.  Since  the  distance  from  a 
to  the  first  horizontal  line  above  ab,  is  one-tenth  of  the 
distance  ad,  it  follows  that  the  distance  measured  on  that 
line  between  ad  and  af  is  one-tenth  of  df:  but  since  one- 
tenth  of  a  tenth  is  a  hundredth,  it  follows  that  this  distance 
is  one  hundredth  (.01)  of  the  unit  of  the  scale.  A  like  dis- 
tance measured  on  the  second  line  will  be  two  hundredths 
(.02)  of  the  unit  of  the  scale ;  on  the  third,  .03  ;  on  tho 
fourth,  .04,  &c. 


SCALES    OF    EQUAL    PARTS.  37 

18.  Hov)  mill  you  take  in  the  dividers  the  unit  of  the  scale 
and  any  number  of  tenths  ? 

Place  one  foot  of  the  dividers  at  1,  and  extend  the  other 
to  that  figure  between  a  and  h  which  designates  the  tenths. 
If  two  or  more  units  are  required,  the  dividers  must  be 
placed  on  a  point  of  division  farther  to  the  left. 

19.  How  do  you  take  off  units,  tenths,  and  hundredths  ? 
Place  one  foot  of  the  dividers  where  the  vertical  line, 

through  the  point  which  designates  the  units,  intersects  the 
line  which  designates  the  hundredths ;  then,  extend  the 
dividers  to  that  line  between  ad  and  be  which  designates 
the  tenths :  the  distance  so  determined  will  be  the  one 
required. 

For  example,  to  take  off  the  distance  2.34,  we  place  one 
foot  of  the  dividers  at  I,  and  extend  the  other  to  e :  and  to 
take  off  the  distance  2.58,  we  place  one  foot  of  the  dividers 
at  p,  and  extend  the  other  io  q. 

Remark  I. — If  a  line  is  so  long  that  the  whole  of  it  can- 
not be  taken  from  the  scale,  it  must  be  divided,  and  the 
parts  of  it  taken  from  the  scale  in  succession. 

Remark  II . — If  a  line  be  given  upon  the  paper,  its  length 
can  be  found  by  taking  it  in  the  dividers  and  applying  it  to 
the  scale. 

20.  How  do  you  construct  a  scale  of  chords  ? 

If  with  any  radius,  as  AC,  we  describe  the  quadrant 
AT),  and  then  divide  it  into  90  equal  parts,  each  part  is 
called  a  degree. 

Through  ^,  and  each  point  of  division,  let  a  chord  be 
drawn,  and  let  the  lengths  of  these  chords  be  accurately 
laid  ofi  on  a  scale  :  such  a  scale  is  called  a  scale  of  chords. 
In  the  figure,  the  chords  are  drawn  for  every  ten  degrees. 

The  scale  of  chords  being  once  constructed,  the  radius 


38 


BOOK    II.— SECTION    I. 


of  the  circle  from  which  the  chords  were  obtained,  is  known ; 
for,  the  chord  marked  60  is  always  equal  to  the  radius  of 


J)^i 

"T^V-^ 

/ 

'^  >^\^^ 

^:^m 

I  i  ! 

/  /  /  /  /^«^ 

qlo    SlO     7[0      C 

0    RO     4.0     3I0     £lo      i\o  ^ 

C 

?                    ^                     A 

the  circle.  A  scale  of  chords  is  generally  laid  down  on 
Gunter's  scale,  and  on  the  scales  which  belong  to  cases  of 
mathematical  instruments,  and  is  marked  oho. 

21.  How  will  you  lay  off  an  angle  with  a  scale  of  chords; 
say  an  angle  of  30  degrees  ? 

Let  AB  be  the  line  from  which  the  angle  is  to  be  laid 
off,  and  A  the  angular  point. 

Take  from  the  scale,  the  chord  of  60 
degrees,  and  with  this  radius  and  the 
point  ^  as  a  Centre,  describe  the  arc 
BC.  Then  take  from  the  scale  the 
chord  of  the  given  angle,  say  30  de- 
grees, and  with  this  line  as  a  radius,  and  I?  as  a  centre, 
describe  an  arc  cutting  BC  in  C.  Through  A  and  C  draw 
the  line  AC,  and  BAG  will  be  the  required  angle. 

22.  Describe  the  semicircular  protractor. 

This  instrument  is  used  to  lay  down,  or  protract  angles. 
It  may  also  be  used  to  measure  angles  included  between 
lines  already  drawn  upon  paper. 

It  consists  of  a  brass  semicircle   ACB,  divided  to  half 


SEMICIRCULAR    PROTRACTOR. 


39 


degrees.     The  degrees  are  numbered  from  0  to  180,  both 
ways ;  that  is,  from  A  to  B,  and  from  B  to  A.     The  di- 


visions, in  the  figure,  are  only  made  to  degrees.  There  is 
a  small  notch  at  the  middle  of  the  diameter  AB,  which  in- 
dicates the  centre  of  the  protractor. 

23.  How  do  you  lay  off  an  angle  with  a  protractor? 
Place  the  diameter  AB  on  the  line,  so  that  the  centre 

shall  fall  on  the  angular  point.  Then  count  the  degrees 
contained  in  the  given  angle  from  A  towards  B,  or  from  B 
towards  A,  and  mark  the  extremity  of  the  arc  with  a  pin. 
Remove  the  protractor,  and  draw  a  line  through  the  point 
so  marked  and  the  angular  point :  this  line  will  make  with 
the  given  line  the  required  angle. 

24.  Describe  gunter's  scale. 

This  is  a  scale  of  two  feet  in  length,  on  the  faces  of 
which,  a  variety  of  scales  are  marked.  The  face  on  which 
the  divisions  of  inches  are  made,  contains,  however,  all  the 
scales  necessary  for  laying  down  lines  and  angles.  These 
are,  the  Scale  of  Equal  Parts,  the  Diagonal  Scale  of  Equal 


40 


BOOK    II. SECTION    I. 


Parts,  and  the   Scale  of  Chords,  all  of  which  have  been 
described. 

25.  How  do  you  bisect  a  given  straight  line ;  that  is,  divide 
it  into  ttDO  equal  parts  ? 

Let  AB  be  the  given  line.  With  A 
as  a  centre,  and  a  radius  greater  than 
half  of  AB,  describe  an  arc  IFE.  Then 
remove  the  foot  of  the  dividers  from  A 
to  5,  and  with  the  same  radius  describe 
the  arc  EHL  Then  join  the  points  / 
and  E  by  the  line  IE:  the  point  Z), 
where  it  intersects  AB,  will  be  the  mid- 
dle of  the  line  AB. 


/; 


A     H\ 


\K 


y 


\ 


V. 


B 


26.  At  a  given  point  in  a  given  straight  line,  how  do  you 
draw  a  perpendicular  to  that  line  ? 

Let  A  be  the  given  point,  and  BC  the  given  line. 

From  A  lay  off  any  two  distances 
AB  and  AC,  equal  to  each  other. 
Then,  from  the  points  B  and  C,  as 
centres,  with  a  radius  greater  than 
BA,  describe  two  arcs  intersecting  ^ 
each  other  in  D:  draw  AD,  and  it 
will  be  the  perpendicular  required. 


n 


A 


SECOND    METHOD. 

27.    When  the  point  A  is  near  the  end  of  the  line. 

Place  one  foot  of  the  dividers  at 
any  point,  as  P,  and  extend  the  other 
leg  to  A.  Then  with  P  as  a  centre, 
and  radius  from  P  to  A,  describe  the 
circumference  of  a  circle.  Through 
C,  where  the  circumference  cuts  BA, 
and  the  centre  P,  draw  the  line  CPD, 


^-^D 


PROBLEMS. 


41 


Then  draw  AD,  and  it  will  be  perpendicular  to  CA,  since 
CAD  is  an  angle  in  a  semicircle. 

28.  Draw  from  a  given  point  without  a  straight  line,  a  per' 
pendicular  to  that  line. 

Let  A  be  the  given  point,  and  BD 
the  given  line. 

From  the  point  J.  as  a  centre,  with 
a  radius  sufficiently  great,  describe  an 
arc  cutting  the  line  BD  in  the  two 
points  B  and  D :  then  mark  the  point 
E,  equally  distant  from  the  points  B  and  D,  and  draw  AE : 
and  AE  will  be  the  perpendicular  required. 


SECOND    METHOD. 

29.  When  the  given  point  A,  is  nearly  opposite  one  end 
of  the  given  line. 

Draw  AC  io  any  point,  as  C  of 
the  line  BD.  Bisect  AC  at  F.  Then 
with  F  as  a  centre,  and  FC  or  FA 
as  a  radius,  describe  the  semicircle 
CD  A.  Then  draw  DA,  and  it  will  ^_ 
be  perpendicular  to  BD  at  D. 

30.  At  a  point,  in  a  given  line,  to  make  an  angle  equal  to 
a  given  angle. 

Let  A  be  the  given  point,  AE 
the  given  line,  and  IKL  the  given 
angle. 

From  the  vertex  K,  as  a  cen- 
tre, with  any  radius,  describe  the  arc  IL,  terminating  in 
the  two  sides  of  the  angle.  From  the  point  A  as  a  centre, 
with  a  distance  AE  equal  to  KI,  describe  the  arc  ED; 
then  take  the  chord  LI,  with  which,  from  the  point  E  as 


42  BOOK    II.  —  SECTION    I. 

a  centre,  describe  an  arc  cutting  the  indefinite  arc  DE,  in 
D:  draw  AD,  and  the  angle  EAD  will  be  equal  to  the 
given  angle  K. 

31.  How  do  you  divide  a  given  angle,  or  a  given  arc,  inU 
two  equal  parts  ? 

Let  C  be  the  given  angle,  and  AEB 
the  arc  which  measures  it. 

From  the  points  A  and  B  as  centres, 
describe  with  the  same  radius  two  arcs 
cutting  each  other  in  D:   through  D  ^^ 

and  the  centre  C  draw  CD :  the  angle 
ACE  will  be  equal  to  the  angle  ECB,  and  the  arc  AE  to 
the  arc  EB. 

32.  How  do  you  draw  through  a  given  point  a  line  pareU 
lei  to  a  given  line  1 

Let  ^  be  the  given  point,  and         r  /; 

BC  the  given  line. 

From  -<4  as  a  centre,  with  a  ra- 
dius greater  than  the  shortest  dis-  ^  "" 
tance  from  A  to  BC,  describe  the  indefinite  arc  ED:  from 
the  point  ^  as  a  centre,  with  the  same  radius,  describe  the 
arc  AF ;  make  ED  =  AF,  and  draw  AD :  then  will  AD 
be  the  parallel  required. 

33.  If  two  angles  of  a  triangle  are  given,  how  do  you  find 
the  third? 

Draw   the    indefinite    line  ^  jj 

DEF.     At  the  point  E,  make  \  / 

the  angle  DEC  equal  to  one  ^\      / 

of  the  given  angles,  and  then     X)  E  F 

the  angle  CEH  equal  to  the 

,  other :  the  remaining  angle  HEF  will  be  the  third  angle 
required. 


PROBLEMS. 


43 


34.  If  two  sides  and  the  included  angle  of  a  triangle  are 
given,  how  do  you  describe  the  triangle  ? 

Let  the  line  5  =  150  feet,  and       „ 
C  =  120  feet,  be  the  given  sides  ; 
and    ^  =  30   degrees,   the    given 
angle :  to  describe  the  triangle  on 
a  scale  of  200  feet  to  the  inch. 

Draw   the   indefinite   line    DG, 
and  at  the  point  I),  make  the  angle 

GDH  equal  to  30  degrees  ;  then  lay  off  DG  equal  to  three 
quarters  of  an  inch,  and  it  will  represent  the  side  B  =  150 
feet :  make  DH  equal  to  six-tenths^  of  an  inch,  and  it  will 
represent  C  =  120  feet :  then  draw  GH,  and  GDH  will  be 
the  required  triangle. 

35.  If  the  three  sides  of  a  triangle  are  given,  how  do  you 
describe  the  triangle? 

Let  A,  B,  and  C,  be  the  sides. 
Draw  DE  equal  to  the  side  A. 
From  the  point  Z)  as  a  centre, 
with  a  radius  equal  to  the  sec- 
ond side  B,  describe  an  arc : 
from  JS  as  a  centre,  with  a  radius 
equal  to  the  third  side  C,  de- 
scribe another  arc  intersecting  the  former  in  F ;  draw  DF 
and  EF,  and  DEF  will  be  the  triangle  required. 

36.  If  two  sides  of  a  triangle  and  an  angle  opposite  one 
of  them  are  given,  how  do  you  describe  the  triangle  ? 

Let  A  and  B  be  the  given 
sides,  and  C  the  given  angle, 
which  we  will  suppose  is  oppo- 
site the  side  B.  Draw  the  in- 
definite line  DF,  and  make  the 
angle   FDE  equal  to  the  angle 


44  BOOK    II. SECTION    I. 

C:  take  DE  =  A,  and  from  the  point  £  as  a  centre,  with 
a  radius  equal  to  the  other  given  side  B,  describe  an  arc 
cutting  DF  in  F ;  draw  EF:  then  will  DEF  be  the  required 
triangle. 

If  the  angle  C  is  acute,  and       j|____ 

the  side  B  less  than  A,  then     j^  ^^^^^ 

-the    arc    described   from   the  „ 

centre    E    with    the    radius 

EF  =  B  will  cut  the  side  DF        ^ 

in  two  points,  F  and  G,  lying  ^N7  ~/^ 

on  the  same  side  of  D  :  hence  *" — ' 

there  will  be   two    triangles, 

DEF  and  DEG,  either  of  which  will  satisfy  all  the  con- 
ditions of  the  problem. 

37.  If  the  adjacent  sides  of  a  parallelogram^  with  the  angle 
which  they  contain,  are  given,  how  do  you  describe  the  paraU 
lelogram  ? 

Let  A  and  B  be    the  given 
sides,  and  C  the  given  angle. 

Draw  the  line  DE  =  A  ;    at  > 

the    point   D,   make    the    angle      ^ 
EDF  =  C ;  take  DF  =  i5 .-  de-     ^^ 
scribe  two  arcs,  the  one  from  F 

as  a  centre,  with  a  radius  FG  =  DE,  the  other  from  E,  as 
a  centre,  with  a  radius  EG  —  DF ;  through  the  point  G, 
where  these  arcs  intersect  each  other,  draw  FG,  EG ;  then 
DEGF  will  be  the  parallelogram  required. 

*  38.  How  do  you  describe  the  circumference  of  a  circle  which 
shall  pass  through  three  given  points  ? 

Let  A,  B,  and  C,  be  the  three  given  points. 
Join  these  points  by  straight  lines  AB,  BC,  CA.     Then 
bisect  any  two  of  these  straight  lines  by  the  perpendicu- 


PROBLEMS. 


45 


lars  OF,  OD,  as  in  Section  25, 
and  the  point  0,  where  these 
perpendiculars  intersect  each 
other,  will  be  the  centre  of  the 
circle. 

Place  one  foot  of  the  dividers 
at  this  centre,  and  extend  the 
other  to  ^,  -B,  or  C,  and  then  with 
this  radius,  let  the  circumference 
be  described. 


XX^ 


39.  How  do  you  find  the  centre  of  a  circle  when  the  circum- 
ference is  given  ? 

Take  any  three  points,  as  A,  B,  and  C,  (see  last  figure,) 
and  join  them  by  the  lines  AB  and  BC.  Then  bisect  these 
lines  by  the  perpendiculars  OD  and  OF,  and  O  will  be  the 
centre  of  the  circle. 


40.  How  do  you  divide  a  given  line  AB,  into  any  number 
of  equal  parts  ? 

Let  AB  be  the  given  line  to  be 
divided.  Let  it  be  required,  if  you 
please,  to  divide  it  into  five  equal 
parts. 

Throughout  A,  one  extremity  of 
the  line,  drawA  h,  making  an  angle 

with  AB.  Then  lay  off  on  Ah,  five  equal  parts,  Ac,  cd,  df 
fg,  gh,  after  which  join  h  and  B.  Through  the  points  of 
division  c,  d,  f,  and  g,  draw  lines  parallel  to  hB,  and  they 
will  divide  AB  into  the  required  number  of  equal  parts. 


4L  How  do  you  describe  a  square  on  a  given  line  ? 
Let  AB  be  the  given  line.     At  the  point  B,  draw  BC 
perpendicular  to  AB,  and  then  make  it  equal  to  BA. 


46 


BOOK    II. SECTION    I. 


Then,  with  A  as  a  centre,  and 
radius  equal  to  AB,  describe  an 
»arc  ;  and  with  C  as  a  centre,  and 
the  same  distance  AB,  describe 
another  arc,  and  through  D,  their 
point  of  intersection,  draw  AD  and 
CD;  then  will  ABCD  be  the  re- 
quired square. 


-~,n! 


41.  How    do   you   construct  a  rhombus,   having  given    the 
length  of  one  of  the  equal  sides  and  one  of  the  angles  ? 

Let  AB  be  equal  to  the  given 
side,  and  E,  the  given  triangle. 

At  B,  lay  off  an  angle  ABC, 
equal  to  E,  and  make  BC  equal 
to  AB.  Then  with  A  and  C  as 
centres,  and  a  radius  equal  to 
AB,  describe  two  arcs,  and 
through  D,  their  point  of  inter- 
section, draw  the  lines  AD  and  CD,  and  ABCD  will  be 
the  required  rhombus. 


42.  How  do  you  inscribe  a  circle  m  a  given  triangle  ? 

Let  ABC  be  the    given 
triangle. 

Bisect  either  two  of  the 
angles,  as  A  and  C,  by  the 
lines  AG  and  CO,  and  the 
point  of  intersection  O  will 
be  the  centre  of  the  in- 
scribed circle.  Then,  through  the  point  of  intersection  O, 
draw  a  line  perpendicular  to  either  side,  and  it  will  be  the 
radius. 


PROBLEMS. 


47 


43.  How  do  you  inscribe  an  equilateral  triangle  in  a  cir- 


-J).- 


cle? 

With  any  point  ^,  as  a  centre,  and 
radius  equal  to  the  radius  of  the  circle, 
describe  an  arc  cutting  the  circum- 
ference in  B  and  C.  Then  bisect  the 
arc  BDC,  after  which,  draw  BC,  BD, 
and  CD,  and  BDC  will  be  an  equi- 
lateral triangle. 

44.  How  do  you  inscribe  a  hexagon  in  a  circle? 
Describe   the   equilateral  triangle 

as  before.     Then  bisect  the  arc  CD 

in  F,  and  the  arc  BD  at  G,  and  draw 

AC,  CF,  FD,  DG,   GB,  and  BA,       I 

and  ACFDGB  will  be  the  hexagon       1 

required.     Or  the   hexagon  may  be  ^  J5 

inscribed  by  applying  the  radius  six 

times  around  the  circumference.  -^ 

45.  How  do  you  inscribe  a  dodecagon  in  a  circle  ? 

Bisect  the  arcs  which  subtend  the  chords  of  the  hexa- 
gon, and  throifgh  the  points  of  bisection  draw  chords,  and 
there  will  be  formed  a  regular  dodecagon. 

46.  How  do  you  inscribe  in  a  circle  a  regular  'pentagon  1 
Draw  the  diameters  AF  and  MN 

at  right  angles  to  each  other,  and 
bisect  the  radius  ON  at  E.  From 
£  as  a  centre,  and  EA  as  a  radius, 
describe  the  arc  As ;  and  from  the 
point  ^  as  a  centre,  and  radius  As, 
describe  the  arc  sB. 

Join  the  points  A  and  B,  and  the 
line  AB,  being  applied  five  times 
around  the  circle,  will  form  the  required  pentagon. 


48 


BOOK    II, SECTION    I. 


47.  How  do  you  inscribe  in  a  circle  a  regular  decagon? 
For  the  decagon,  bisect  the  arcs  which  subtend  the  sides 

of  the  pentagon,  and  join  the  points  of  bisection ;  and  the 
lines  so  drawn  will  form  the  regular  decagon. 

48.  How  will  you  inscribe  in  a  circle  a  polygon  having  any 
number  of  sides  ? 

Divide  the  circumference  of  the  circle  into  as  many  equal 
parts  as  there  are  sides  of  the  polygon,  and  draw  lines 
through  the  points  of  division :  these  lines  will  be  the  sides 
of  the  required  polygon. 

49.  How  do  you  inscribe  a  square 
in  a  given  circle  ? 

Let  ABCD  be  the  given  circle. 
Draw  two  diameters  DB  and  AC 
at  right  angles  to  ea^h  other,  and  ^1 
through  the  points  A,  B,  C,  and 
D,  draw  the  lines  AB,  BC,  CD, 
and  DA:  then  ^^ CD  will  be  an 
inscribed  square. 

50.  How  would  you  inscribe  an  octagon  ? 

By  bisecting  the  arcs  AB,  BC,  CD,  and  DA,  and  join- 
ing the  points  of  bisection,  we  can  form  an  octagon ;  and 
by  bisecting  the  arcs  which  subtend  the  sides  of  the  octa- 
gon, we  can  inscribe  a  polygon  of  sixteen  sides. 

51.  How  will  you  circumscribe  a  square  about  a  circle^ 
Draw  two  diameters  AB  and  CD 

at  right  angles  to  each  other;  and 
through  their  extremities  A,B,C,  and 
D,  draw  lines  respectively  parallel  to 
the  diameters  CD  and  AB :  a  square 
will  thus  be  formed  circumscribing 
the  circle. 


D 

1 

A 

K 

J 

C 

B 


PROBLEMS. 


49 


52.  How  do  you  draw  a  line  ^  which  shall  he  tangent  to  the 
circumference  of  a  circle  at  a  given 
point  ? 

Let  A  be  the  given  point- 
Through  -4  draw  the  radius  AC, 
and  then  draw  DA  perpendicular 
to  the  radius  at  the  extremity  A. 
The  hne  DA  will  be  tangent  to 
the  circumference  at  the  point  A. 


53.  How    do  you   draw   through   a  given  point  without 
circle  a  line  which  shall  he  tangent  to  the  circumference? 

Let  A  be  the  given  point  without 
the  given  circle  BED.  .  Join  the 
centre  C  and  the  given  point  A^ 
and  bisect  tlie  line  CA  at  O. 

With  O  as  a  centre,  and  OA  as 
a  radius,  describe  the  circumfer- 
ence ABCD.  Through  B  and  D 
draw  the  lines  AB  and^Z),  and 
they  will  be  tangent  to  the  circle 
BED  at  the  points  B  and  D. 


54.   What  is  an  ellipse  ? 

It  is  an  oval  curve  ACBD, 


55.  W^  is  the  longest  line  wl^ich  can  he  drawn  within 
tfle  curve  called,?      What  is  the  shortest  line  called? 

The  longest  line  AB  is  called  the  transverse  axw;  and 
the  shortest  line  DC  is  called  the  conjugate  axis.  The 
point  E,  at  which  they  intersect,  is  called  the  centre  of  the 
ellipse.  .  .  V 

3 


50 


BOOK    II. SECTION    1. 


56.  What  are  the  foci  of 
an  ellipse  1 

They  are  two  points  F 
and  i7,  determined  by  de- 
scribing the  arc  of  a  circle 
with  D  as  a  centre,  and 
a  radius  BF  equal  to  AE^ 
half  of  the  transverse  axis. 

57.  Flow  will  you  describe  an  ellipse  when  you  knotd  the 
two  axes  AB  and  CD  ? 

First,  find  the  foci  F  and  H  by  describing  an  arc  with 
D  as  a  centre,  and  with  a  radius  equal  to  AE. 

Secondly,  take  a  string  or  thread  equal  in  length  to  ABy 
and  fasten  the  extremities  at  the  foci  F  and  H.  Then 
place  a  pencil  a,gainst  the  string  and  move  it  round,  bear- 
ing it  tight  against  the  string,  and  the  point  will  describe 
the  ellipse  ADBC, 


QUESTIONS  TO  BE  PUT  FROM  FIGURES  MADE  BY  THE  TEACHER 
UPON  THE  BLACK-BOARD. 

SECTION    I. 


What  is  a  line  ? 

What  is  a  right  line  T 

What  is  a  curve  ? 

What  does  the  word  line  imply  ? 

What  is  a  surface  ? 

What  is  a  plane  7 

What  are  parallel  right  lines^ 

What  are  parallel  curves'? 

What  are  oblique  lines? 

What  are  horizontal  lines? 

What  are  vertical  lines? 

What  is  an  angle?  how  readT 

What  are  curvilinear  angles? 

When  is  one  line  perpendicular  to 

another? 
What  are  the  angles  then  called? 


What  is  an  acute  angle  ? 

What  is  an  obtuse  angle? 

What  follows  when  two  lines  inter- 
sect eacb  other  ? 

What  fdlows  when  on©  line  cuts 
two  parallels? 

What  follows  when  one  line  is  per- 
pendicular to  one  of  several  paral- 

.   lels? 

How  many  lines  ceoi  be  drawn  from 
a  pqint  perpen^cular  to  a  given 
line? 

If  oblique  lines  are  drawn,  how  do 
they  compare  ? 

What  is  the  circumfere;nce  of  a  cir- 
cle? . 


QUESTIONS. 


51 


For  what  is  it  used  ? 

How  is  it  divided  ? 

How  are  angles  measured? 

How  many  degrees  in  one  right  an- 
gle? 

What  is  one  quarter  of  the  circum- 
ference called  ?    One  half? 


When  one  straight  lino  meets  an- 
other, what  is  the  sum  of  the 
angles  on  the  same  side  ? 

If  there  are  several  angles,  what  is 
their  sum  equal  to? 

What  is  the  sum  of  all  the  angles 
about  a  given  point  equal  to? 


SECTION    II. 


What  is  a  plane  figure  ? 

What  is  it  called  when  the  bound- 
ing lines  are  straight  ? 

What  are  the  lines  themselves 
called? 

What  i^  a  triangle  ? 

What  is  a  quadrilateral  ? 

What  is  a  polygon  of  five  sides? 

What  is  a  polygon  of  six  sides  ? 

What  is  a  polygon  of  seven  sides? 

What  is  a  polygon  of  eight  sides  ? 

What  is  a  polygon  of  nine  sides  ? 

What  is  a  polygon  of  ten  sides  ? 

"^hat  is  a  polygon  of  twelve  sides  ? 


What  is  the  smallest  number  of 
straight  hues  which  can  enclose 
a  space  ? 

What  are  the  several  kinds  of  tri- 
angles ? 

What  is  the  base  of  a  triangle  ? 

What  its  altitude? 

What  are  the  different  kinds  of 
quadrilaterals? 

What  is  the  base  of  a  figure  ? 

What  is  a  diagonal  ? 

What  is  the  square  described  on  the 
hypothenuse  of  a  right-angled  tri- 
angle equal  to  ? 


SECTION 

What  is  a  circle  ? 

What  is  a  circumference  ? 

What  is  the  radius  of  a  circle  ? 

What  is  a^  arc  ? 

What  is  a  chord  ? 

What  is  a  segment? 

What  is  a  sector-?     • . 

Wliat  is  an  angle  at  the  centre? 

What  is  an  angle  at  the  circum- 
ference ? 

W|iat  is  an  angle  in  a  segment  ? 

What  is  a  secant  line  ? 

What  is  a  tangent  line? 

What  position  has  the  tangent  with 
the  radius  ? 


Ill, 

When  is  a  figure  said  to  be  inscribed 
in  a  circle? 

When  circumscribed  about  it  ? 

How  is  an  angle  at  the  centre  of  a 
circle  measured  ? 

What  measures  an  angle  at  the  cir- 
cumference ? 

What  is  the  sum  of  the  three  angles 
of  a  triangle  equal  to  ? 

How  does  a  perpendicular  throagh 
the  centi-e  divide  the  chord  ? 

How  do  the  distances  from  the  cen- 
tre to  equal  chords  cofmpare  with 
eaclr  other  ? 


52 


BOOK    II.— SECTION    I. 


PRACTICAL    GEOMETRY. 


What  is  Practical  Geometry  ? 

What  is  a  problem  ? 

What  are  the  dividers? 

How  do  you  lay  off  a  line? 

How  do  you  describe  the  circum- 
ference of  a  circle  ? 

How  on  the  black-board  ? 

Describe  the  ruler  and  triangle,  and 
the  manner  of  using  them. 

How  do  you  draw  a  perpendicular  ? 

What  is  a  Scale  of  Equal  Parts  ? 

What  is  a  unit  of  the  scale  ? 

Explain  how  you  take  from  the 
scale  a  given  number  of  parts. 

Explain  the  Diagonal  Scale. 

What  is  a  Scale  of  Chords? 

How  will  you  lay  off  an  angle  ? 

What  is  the  Semicircular  Protrac- 
tor? 

How  do  you  lay  off  an  angle  with  it  ? 

Describe  Gunter's  Scale. 

Plow  do  you  bisect  a  line  ? 

How  do  you  draw  a  perpendicular 
at  a  given  point  ? 

How  do  you  make  an  angle  equal 
to  a  given  a^igle  ? 

How  do  you  bisect  an  arc  ? 

How  do  you  draw  a  parallel  to  a 
given  line?    . 

When  two  angles  of  a  triangle  are 
given,  how  do  you  find  the  third  ? 


When  two  sides  and  the  included 
angle  are  given,  how  do  you  de- 
scribe the  triangle  ? 

How  do  you  describe  a  parallelo- 
gram with  the  same  given  ? 

How  do  you  pass  the  circumference 
of  a  circle  through  three  points  ? 

How  do  you  divide  a  line  into  any 
number  of  equal  parts  ? 

How  do  you  describe  a  square  ? 

How  do  you  construct  a  rhombus  ? 

Hov/  do  you  inscribe  a  circle  in  a 

,   given  triangle  ? 

How  do  you  inscribe  an  equilateral 
triangle  in  a  given  circle  ? 

How  do  you  inscribe  a  hexagon  in 
a  circle  ? 

How  do  you  inscribe  a  dodeca- 
gon? 

How  do  you  inscribe  in  a  circle  a 
polygon  having  any  number  of 


How  do  you  inscribe  a  square  ?  an 
octagon  ? 

How  do  you  circumscribe  a  square 
about  a  circle  ? 

How  do  you  draw  a  line  tangent  to 
a  circle  at  a  point  of  the  circum- 
ference ? 

How  from  a  point  without  the  cir- 
cumference ? 


Note. — After  the  teacher  shall  have  made  the  above  figures,  or  most 
of  t}iem,"on  the  black-board,  and  the  pupils  copied  them  on  their  slates, 
let  the  students  then  be  called  to  the  black-board  in  turn^  and  practised 
in  the  drawing  of  them. 


OF    DRAWING    IN    GENKRAL.  63 


BOOK  III. 

SECTION   I. 

OF    DRAWING    IN    GENERAL. 

1;    What  are  drawings? 

Drawings  are  representations  to  the  eye  of  the  forms, 
dimensions,  positions,  and  appearance  of  objects.  They 
form  a  written  language,  which  is  easily  comprehended  by 
every  one. 

2.  What  are  the  uses  of  drawing? 

Drawing,  to  the  practical  man,  furnishes  a  simple  means 
of  describing  and  explaining  a  thing  in  a  brief  and  striking 
manner.  On  this  account,  alone,  its  great  advantages  are 
everywhere  apparent.  Drawings,  also,  impress  the  mind 
with  images  approaching  nearer  to  the  reality,  than  any 
other  means  of  description.  The  pen  of  the  ablest  historian 
presents  but  a  feeble  image,  when  compared  with  the  pic- 
tured canvass  of  the  painter,  or  the  life-like  forms  of  the 
sculptor. 

3.  When  you  look  at  a  single  object,  what  do  you  observe 
that  distinguishes'  it  from  other  objects  ? 

When  we  observe  a  single  object,  we  discover  that  we 
are  able  to  recognise  it  by  means  of  three  .properties  which 
distinguish  it  from  other  objects,  viz.:  its'  form,  its  light 
and  shade,  and  its  color.      If  we    consider  more   than  one 


54 


BOOK    III. SECTION    I. 


object,  we   are   then  able  to  distinguish  them  from  each 
other,  by  their  relative  position,  also. 

4.  How  do  you  illustrate  the  idea  of  form.? 

If  we  join  any  three  points 
A,  B,  and  C,  by  straight 
lines,  the  result  will  he  a,  figure 
or  form  of  a  triangle  i  If  we 
take  another  point  D,  and  join 
the  three  points  A,  D^.  and  C, 
we  shall  have  the  form  of  an- 
other triangle  ADC.  The 
straight  lines  which  bound 
each  of  these  figures,  make 
up  what  is  called  its  out- 
line. ^ 

If  with  C  as  a  centre,  and  any  radius,  we  describe  the 
circumference  of  a  circle,  the  curve 
so  drawn  will  be  the  outline  of  the 
circle. 

Now,  the  triangle  and  circle  differ 
from  each  other  only  in  form,  and 
ithe  -form  is  determined  by  the  out- 
line': hence  we  see  that  outline  is 
one  means  of  representing  form  to 
the  eye.  It.  is  thus  that  we  are 
-able  to  distinguish  a  triangle  from  a  circle,  and  a  circle 
from  a  square ;  and  the  drawings  of  their  outlines  present 
to  the  mind,  through  the  eye,  the  idea  of  the  objects  them- 
selves. 


5.  How  do  you  illustrate  light  and  shade  I 
If  we  hold  any  object  in  the  sun's  rays,  it  is  evident,  that 
that  part  of  it  which  is   turnerd   towards   the   sUn  will  be 
lighted;  and  that  the  part  which  is  turned  away  from  the 


OF    DRAWING    IN    GENERAL. 


65 


sun  will  be  comparatively  dark.     The  part  towards  the  sun 
is  called  the  light;  the  other  part,  the  shade. 


6.  In  what  manner  do  light  and  shade  modify  the  idea  of 
a  form  which  is  represented  only  by  its  outline  ? 

The  circle  whose  centre  is  C, 
is  the  outline  of  so  much  of  the 
flat  white  paper  as  is  contained 
within  its  circumference . 

Now,  if  we  observe  a  sphere,  or 
perfectly  round  ball,  we  find  that, 
in  every  position,  its  outline  is  also 
a  circle.  We  cannot  tell,  there- 
fore, whether  this  circle  is  the  out- 
line of  a  circular  piece  of  paper  or  of  a  sphere. 


Let  the  circle  whose  centre  is 
D,  be  the  outline- of  a  sphere.  If 
we  suppose  the  light  to  proceed 
from  the  left  hand,  then  the  part 
of  the  sphere  towards  the  left  will 
bfe  the  light,  and  the  part  towards 
the  right,  the  shade. 


Leaving  the  white  paper  for  the 
light,  we  will  represent  the  shade, 
or  dark  part,,  by  means  of  lines 
drawn  in  such  a  manner,  as  to 
darken  that  part  occupied  by  the 
shade. 


56 


BOOK    III. SECTION    I. 


In  a  similar  manner, 
the  outline  of  a  rectangle 
may  be  distinguished  from 
that  of  a  cylinder  by  means 
of  light  and  shade.  Thus 
we  see  that  light  and  shade 
furnish  a  distinction  be- 
tween objects  whose  outlines  are  the  same. 


Rectangle 

1 

1 

Cylinder 


7.  In  how  many  ways  may  the  shade  on  a  body  be  modi- 
fed? 

In  two  ways :  viz ,  in  its  depth  or  intensity,  and  its 
color. 

8.  How  do  you  know  which  part  of  a  body  has  the  greatest 
depth  or  intensity  of  shade  ? 

If  there  were  no  atmosphere,  and  no  body  in  existence 
except  the  one  we  are  considering,  that  part  of  it  which 
does  not  receive  the  sun's  rays  would  be  invisible.  But 
since  the  atmosphere,  as  well  as  every  other  substance  in 
nature,  reflects  back  the  light  which  it  receives,  casting  it 
in  a  direction  contrar)^  to  that  of  the  sun's  rays  ;  it  follows, 
that  the  part  of  any  object  which  does  not  receive  tlie  direct 
light  of  the  sun,  will  yet  receive  light  from  other  objects, 
behind  it  with  reference  to  the  sun,  and  will  be  sufficiently 
illuminated  to  exhibit  its  form.  Now,  since  bodies  are  more 
or  less  illuminated  as  they  receive  the  light  directly  or  ob- 
liquely, it  follows,  that  if  we  conceiv-e  the  reflecting  body 
to  be  placed  directly  behind  the  one  receiving  the  light,  that 
the  part  nearest  the  reflecting  body  will  receive  more  light 
than  the  parts  more  remote  ;  and  hence,  the  shade  there 
will  be  less  intense.  It  therefore  follows,  that  the  efl'ect 
of  reflected  light  on  the  depth  of  shade,  will  be  thfe  greatest 
near  the  outline  of  the  body  which  is  farthest  from  the 
source  of  light. 


OF    DRAWING    IN    GENERAL. 


57 


9.  How  may  the  shade  of  an  object  be  modified  in  regard 
to  color? 

Every  reflected  ray  of  light  is  of  the  same  color  as  the 
body  which  reflects  it,  and  when  such  rays  illuminate  a  dark 
object,  they  also  impart  to  it  their  color.  This  may  be 
shown  by  holding  any  dark  body,  as  a  sphere,  in  the  sun's 
rays,  and  placing  near  it,  and  opposite  to  the  sun,  a  piece  of 
bright-colored  red  or  yellow  paper.  The  reflected  rays  from 
the  paper  will  impart  their  tint  to  the  shade  of  the  sphere. 

10.  What  is  the  shadow  on  a  body? 

The  shadow  on  a  body  is  that  part  of  it  from  which  the 
light  is  intercepted  by  some  opaque  body. 

11.  How  may  the  forms  of  objects  be 
discovered  by  means  of  the  shadows  which 
they  cast  or  receive  ? 

It  is  evident  that  the  shadow  of  a  tri- 
angle, or  of  a  square,  on  a  flat  surface, 
will,  in  certain  positions,  exactly  resemble 
the  bodies  which  cast  them.  But  the  sur- 
face which  receives  the  shadow  will  modi- 
fy the  shape  of'  it ;  and  thus  the  shadow 
will  also  give  an  idea  of  the  form  of  the 
surface  on  which  it  falls. 

For  example,  the  rectangle 
in  the  figure  casts  a  shadow 
of  such  a  shape  on  the  wall 
and  step  which  are  behind  it, 
as   to    show   their    form   dis- 
tinctly.    Without  the  shadow, 
the  two  lines  which  are  the    - 
outlines  ,  of    the    step,    might    - 
equally  well  represent  two  Jio- 
rizontal  lines  drawn  upon  the  wall. 
3* 


>8 


BOOK    III. SECTION    I. 


This  example  exhibits  the 
outline,  light  and  shade,  reflec- 
tion, and  shadow  of  a  cup ; 
and  is  an  illustration  of  the 
foregoing  principles. 

12.  How  may  the  relative  position  of  objects  he  determined 
by  the  shadows  which  they  cast  or  receive  ? 

When  a  shadow  is  entirely- 
separated  from  the  body  which 
casts  it,  as  is  the  shadow  of  the 
sphere  in  this  example,  it  is  then 
plain  that  a  space  intervenes  be- 
tween the  body  and  the  surface 
on  which  the  shadow  falls. 

But  when  the  shadow  joins  the  body  which  casts  it,  as 
in  this  example,  then  the  body 
casting  the  shadow  touches  the 
surface    on   which   the    shadow 
falls. 

'Hence,  the  nearer  an  object  is 
to  the  surface  on  which  the  shad- 
ow falls,  the  nearer  will  the  shadow  approach  to  the  object. 


OF    DRAWING    IN    GENERAL. 


59 


The  example  of  the  house  shows,  by  the  shadows  on  B 
and  C,  that  B  stands  further  back  than  A,  and  C  farther 
than  B. 

The  shadows  in  the  example  which  follows,  exhibit  the 
difference  between  the  forms  of  three  objects  whose  out- 
lines are  exactly  the  same.  The  shade  on  them  cannot  be 
repjese^ited  in  these  outlines. 


13,  W^t  may  be  said  of  color,  as  a  means  of  distinguish- 
ing objects  from,  each  other  ? 

Of  this,  it  is  only  necessary  to  observe,  that  when  we 
have  represented  the  form  of  an  object,  its  light  and  shade, 
and  its  shadow,  if  we  wish  still  further  to  distinguish  it  from 
other  objects,  we  have  but  to  add  its  appropriate  color.  For 
example,  in  the  drawing  of  a  machine,  if  we  wish  to  ex- 
hibit the  difference  between  the  wood,  the  iron,  and  the 
brass,  the  natural  colors  of  these  should  be  added  in  the 
drawing. 


14.    What  effect  have  shade  and  shadow? 

Shade  and  shadow  have  the  effect  of  obscuring  the  out- 
line, form,  and  color,  of  that  part  of  every  object  on  which 
they  are  found.  Hence  shading,  in  drawing,  is  the  ob 
souring,  in  imitation  of  nature,  of  those  portions  of  the 
objects  we  are  representing,  and  from  which  the  light  is 
intercepted.  There  is  this  difference,  however,  between 
nature   and  art:— in  the  former  we  distinguish  and  deter- 


60 


BOOK    III. — SECTION    I. 


mine  forms  by  means  of  the  light ;    in  the  latter,  by  the 
shade  and  shadow. 

15.  By  what  is  the  process  of  shading  regulated? 

The  process  of  shading  a  drawing  varies  according  to 
the  instrument  used.  The  pen  is  capable  of  making  only 
lines  and  dots ;  hence,  if  we  employ  it  only,  we  are  con- 
fined to  those  two  methods  of  shading.  The  briish  and 
lead  pencil  possess,  in  addition  to  the  resources  of  the  pen, 
the  capability  of  laying  a  smooth,  graduated  tint  of  shade, 
which  by  the  brush  may  also  be  made  of  any  color  that 
may  be  desired. 


16.   What  may  be  said  of  the  use  of  the  pencil  ? 

The  acquisition  of  a  skilful  and  easy  manner  of  hand- 
ling the  pencil,  depends  in  a  great  measure  upon  the  way 
of  holding  it.  The  thumb,  with  the  first  and  second  fingers, 
should  grasp  the  pencil 
about  an  inch  from  its 
point.  The  thumb  should 
not  be  drawn  back,  as 
we  are  taught  in  holding 
a  pen  for  writing ;  but 
should  be  placed  opposite 
to,  or  a  little  below  the  points  of  the  fingers. 

This  position  will  enable  the  hand 
to  move  from  left  to  right,  and  to 
draw  curved  lines  with  as  much  free- 
dom in  that  direction,  as  from  right  to 
left. 

Let  the  learner  now  practise  the 
drawing  of  such  lines  as  are  shown 
in  the  figure,  from  left  to  right. 

In  drawing  straight  lines  by  the  hand,  the  learner  should 


OF    DRAWING    IN    GENERAL. 


61 


^c^ 


■  0"-^.^o/ 


^i^. 


not  begin  by  tiinidly  drawing 
dotted  lines,  as  is  usually 
done  ;  but  the  pencil  should 
be  passed  rapidly  two  or  three 
times  from  one  >  extremity  of 
the  line  to  the  other,  without 
touching  the  paper,  and  then 
the  line  should  be  drawn  at 
one  stroke.  Should  it  not  be 
correct,  repeat  the  trial  until 
it  is  right;  after  which,  and 
not  before,  efface  whatever 
is  wrong. 

In  the  same  manner, 
curved  lines  may  be  first 
sketched  out  by  drawing 
broken  lines,  and  after- 
wards rounding  off  the  angles  and  effacing  the  straight  lines. 
These  distinctions  may  appear  trifling,  and  too  minute,  but 
nothing  is  more  certain  than  that  a  careful  and  intelligent 
observance  of  them,  will  ensure  a  rapid  and  easy  manner 
of  sketching. 


OENfORAL    REMARKS. 


It  is^not  intended,  nor  would  it  be  possible,  to  give  here 
jnore  than  a  few  practical  hints  concerning  the  general  prin- 
ciples of  the  art  of  drawing.  The  learner,  aftei:  familiar- 
izing himself  with  them,  and  with  the  short  directions  as 
to  the  mechanical  part,  should  copy  some  good  drawings, 
under  the  direction  of  an  instructor.  He  should  then  take 
some  simple  object,  such  as  a  book,  a  cup,  an  inkstand, 
&;c.,  and  placing  it  before  him,  endeavor  to  describe  its  po- 
sition and  proportions  by  means  of  its  outline.  This  is  done 
by  comparing  the  lines  which  -make  up  its  outline  with  each 


62 


BOOK    III. SECTION    I. 


nrlm/mllilffffinfflfflS      Hf^fflffl^^^^^^ 


Other,  regarding  both  their  comparative  length  and  the  an- 
gles which,  they  make  with  each  other.  If  the  direction 
and  relative  length  of  each  line  are  right,  the  drawing  must 
be  correct.  ' 

An  easy  help  in  finding 
the  direction  of  a  line  nearly 
vertical,  is  to  hold  at  arm's 
length,  between  the  eye  and 
object,  (a  pyramid,  for  ex- 
ample,) a  ruler  which  serves 
as  a  plumb  line.  The  edge 
of,  the  ruler  being  vertical, 
when  brought  in  range  with 
the  point  A,  will  show  how 
much  the  line  AC  varies 
from  a  perpendicular. 

Now  by  drawing,  or  imagining  to  be  drawn,  a  vertical 
line  upon  the  paper,  and  then  drawing  a  line  making  with 
it  an  angle  equal  to  BAC,  we  shall  have  the  direction  of 
AC,  or  its  inclination  to  a  plumb-line  AB. 

To  find  the  direction  of  a 
line  nearly  horizontal,  we 
have  but  to  balance  the  ru- 
ler^ by  placing  its  centre 
upon  the  thumb  ;  then,  con- 
tinuing it  in  a  horizontal 
position,  and  bringing  it  to 
range  with  the  point  ^,  we 
discover  how  liiuch  AB  va- 
ries from  a  true  level.  Con- 
ceiving or  drawing  such  an 
auxiliary  level  line  upon  the  paper,  and  then  laying  down 
the  angle  CAB,  we  shall  have  the  direction  of  AB,  or  its 
inclination  to  a  horizontal  line.     This  method  is  applicable 


TOPOGRAPHICAL    DRAWING.  63 

to  the  lines  of  distant  objects,  as  well  as  to  those  which 
are  near. 

Having  acquired  by  practice  the  power  of  sketching  a 
single  object  in  outline,  the  learner  should  place  two  or 
more  objects  before  him,  and  endeavor,  by  means  of  draw- 
ing their  outlines,  to  represent,  in  addition  to  their  forms, 
their  relative  position  with  respect  to  each  othef.  He  should 
then  proceed  to  shade  them,  and  to  draw  the  shadows  which 
they  cast  upon  each  other,  and  upon  the  table  or  other  sur- 
face on  which  they  may  be  placed.  The  colors  of  the 
lights,  shades,  and  shadows  may  then  be  added,  and  the 
representation,  will  be  complete. 


SIECTION  11. 

TOPOGRAPHICAL    DRAWING. 

1 .   What  is  Topographical  Drawing  ? 

Topographical  Drawing  is  the  art  of  representing  upon 
a  plane  surface,  the  character  and  features  of  any  piece 
of  ground.  Such  drawings  are  always  plans,  and  are  dis- 
tinguished from  geographical  maps  by  a  greater  degree  of 
minuteness  in  their  details.  A  system  of  signs  has  been 
universally  agreed  upori^  and  adopted;  most  of  which,  how- 
ever, have  a  sufficient  resemblance  to  the  objects  for  which 
they  stand,  to  enable  them  to  be  easily  recognised. 

The  signs  in  the  aimexed  plates  have  been  adopted  by 
the  Engineer  Department,  and  are  used  in  all  the  plans 
and  maps  made  by  the  U.  S.  Engineers. 

These  we  shall  proceed  to  explain,  giving  at  the  same 
time  such  hints  as  to  the  manner  of  drawing  tliem,  as  may 
appear  to  be  necessary.  . 


64  BOOK    III. — SECTION    II. 

The  dimensions  in  which  we  represent  such  objects  as 
houses,  trees,  roads,  &;c.,  in  a  topogra;phical  plan,  depend, 
of  course,  upon  the  scale  to  which  the  drawing  is  made. 
Generally,  for  the  sake  of  greater  distinctness,  they  are  en- 
larged to  two  or  three  times  their  proportionate  size  :  un- 
less the  scale  is  very  large,  or  when  one  of  the  objects  of 
the  plan  is  to  exhibit  every  thing  in  its  just  proportion. 

2.  Explain  the  figures  on  the  next  page. 

The  figures  in  the  first  column  explain  themselves,  in 
most  cases,  by  some  resemblance' or  appropriate  sign;  in 
other  cases,  they  are  purely  conventional. 

In  Fig.  2,  the  signs  of  the  plants  are  placed  on  the  cor- 
ners of  squares  drawn  through  the  fields  they  occupy. 

Fig.  3  shows  the  manner  of  expressing  a  pine  forest  with 
roads  and  the  details^  of  the  leaves,  in  case  the  scale  of  the 
drawing  will  admit  of  their  use.  In  forests,  the  trees  are 
placed  without  any  particular  order  or  arrangement. 

In  Fig.  4,  the  horizontal  lines,  or  the  lines  parallel  to  the 
top  and  bottom  of  the  drawing,  represent  the  watery  portion 
of  a  fresh-water  marsh :  the  rest  of  the  figure,  the  earthy  or 
grassy  parts.  In  general,  stagnant  water  is  represented  by 
horizontal  lines ;  and  meadow,  or  heath,  by  small  tufts  of 
grass.  The  combination  of  these  two  signs  indicates  mo- 
rass, or  wet  ground. 

Fig.  5  represents  hillocks,  or  sloping  ground.  The  paper 
is  always  left  white  to  denote  a  level;  and  each  one  of 
the  broken  lines  drawn  from  the  summit  to  the  base  of  a 
hill,  indicates  throughout  its  length  t^e  direction  of  the  slope, 
or  the  line  of  greatest  descent.  The  degree  of  blackness, 
or  shade,  produced  by  these  lines  shows  the.  nature  of  the 
slope,  from  the  perfect  white  of  a  level,  to  the  deep  black- 
ness of  an  almost  perpendicular  descent. 


f)C)  BOOK    III. SECTION    II. 

3.  Explain  the  figures  on  the  next  page. 

Fig.  1  represents  a  rice-plantation ;  Fig.  2,  an  ornamen 
tal  garden ;  Fig.  3,  a  cotton-field ;  Fig.  4,  ploughed  land , 
Fig.  5,  an  orchard;  and  Fig.  6,  a  vineyard.  Figs.  1,  3,  5, 
and  6,  are  drawn  as  was  described  in  the  case  of  page  65, 
Fig.  2.  Where  it  is  not  necessary  to  describe  minutely  the 
kind  of  crop  existing  upon  the  land,  every  kind  of  cultiva- 
tion may  be  expressed  as  is  done  in  Fig.  4. 

Figs.  7,  8,  and  9  indicate,  respectively,  the  details  of  the 
leaves  for  oak,  fruit,  and  chestnut  trees,  whenever  their  use 
in  a  plan  is  desirable. 

Fig.  10  represents  a  heath  and  common  road.  It  is  left 
white,  being  a  level,  with  the  exception  of  the  tufts  of 
grass. 

Fig.  11  is  an  oak,  &c.  forest. 

Fig.  12  is  a  salt  marsh.  This  is  drawn  in  a  different 
manner  from  a  fresh-water  marsh,  being  composed  of  un- 
broken horizontal  lines,  with  tufts  of  grass  interspersed 
among  them. 

Fig.  13  represents  meadow,  or  bottom  land,  with  a  small 
stream  running  through  it.  The  sign  for  the  grass  is  here 
more  regularly  disposed  than  in  a  heath,  or  common. 

Fig.  14  shows  the  mode  of  indicating  different  kinds  of 
roads,  fences,  paths,  &c. 

4.  How  is  water  represented? 

Running  watet,  the  water  of  lakes,  and  water  that  is 
affected  by  tides,  are  always  represented  by  lines  drawn 
within  the  outline,  and  parallel  to  the  shores,  in  such  a 
manner,  that  by  gradually  increasing  the  distance  between 
the  lines,  which  are  at  first  very  close  together,  the  shade 
may  be  uniformly  lightened  from  the  shores  to  the  middle. 
The  course  of  the  current  is  indicated  by  an  arrow,  with 
the  head  turned  in  the  direction  in  which  the  water  runs. 


Jig:.l: 


Fiff.  4. 


ill 

i 


Fig.  7. 


Fik-  2. 


Fig.  5. 

4,  ^  ^  %  ^  <\  "^ 

4  <^.  ^  \  %  ^  4^ 
^  ^  (^  ,§jj,  <sj^^  ^  4^ 

^  ^  ^^  ^  %  i, 

Fig.  8. 


Fig.3. 

Fig.  6. 


%  s 


Fig.  10. 


Fig.  11. 


Fig.  12. 


Fig.  13. 


BBSnaiiiii! 


(Fig.  14.) 


iiiiiiiliiiiiliWIir;! 


(f  ig.  14. ;  - 

JillilliMiariliagalJmS  \AAA/\    Woo.Vn  fence     AA^'^A/ 

path 


68  BOOK    III. SECTION    II. 

5,  Explain  the  figures  on  the  next  page. 

Fig.  1  represents  the  rocky  shore  oi  water  thus  shaded. 

Fig.  2  denotes  rocks  that  are  above  the  surface  of  the 
water.  Here,  also,  the  lines  indicate  the  direction  of  the 
descent  from  the  highest  point,  near  the  middle,  to  the 
water  line. 

Fig.  3  shows  the  manner  of  representing  salt-works. 

Figs.  4,  5,  and  6  show  the  three  conditions  of  sand-shoals. 

Fig.  7  is  a  sign  used  to  show  the  direction  of  the  current. 

Fig.  S  shows  that  there  is  no  current. 

Fig.  9  indicates  the  different  stages  of  the  tides  by- 
means  of  dots  introduced  in  all  shading  above  low-water 
mark. 

Fig.  10  represents  rocks  sometimes  hare,  and  Fig.  11, 
sunken  rocks. 

Fig.  12  is  a  shore  with  sand-hillocks  and  fisheries. 

Fig.  13  is  a  collection  of  signs  used  for  describing  the 
facilities  or  dangers  of  Navigation. 

Fig.  14  exhibits  a  river,  with  the  different  circumstances 
connected  with  its  navigation,  and  the  means  of  crossing  it. 

Fig.  15  is  a  ^ake,  shaded  in  the  manner  before  de- 
scribed. 

In  shading  a  piece  of  water  by  this  method,  this  rule  must 
be  observed.  Having  drawn  the  outline,  conduct  the  first 
shading  line  along  every  shore,  (if  there  be  more  than  one,) 
and  around  all. islands,  keeping  it  as  close  as  possible  to 
the  shore -line. 

When  the  first  shading  line  is  thus  applied -every where, 
take  up  the  second  one,  laying  it  nearly  as  close  to  the 
first  as  the  first  is  to  the  outline.  When  the  second!  hne  is 
drawn  wherever  it  can  go,  take  up  the  third ;  increasing 
gradually  and  uniformly  the  distance  between  the  lines,  un- 


toiiWiiii>A.A>:\vA;;////////^^5^-'^i 


70  BOOK    III. SECTION    III. 

til  they  approach  the  middle,  when  it  may  be  increased  a 
little  more  rapidly,  and  the  lines  made  somewhat  thinner. 

By  pursuing  this  system,  the  shade  will  be  graduated  in 
a  similar  manner  from  every  shore,  and  perfect  symmetry 
in  the  positions  of  the  lines  will  be  insured. 


SECTION   III. 

PRINCIPLES    OF    PLAN    DRAWING. 

1.  What  are  Geometrical  Drawings? 

Geometrical  drawings  are  those  which  are  made  for  the 
purpose  of  conveying  to  the  mind,  through  the  eye,  a  just 
idea  of  the  true  proportions  and  dimensions  of  objects. 

2.  What  objects  are  generally  represented  in  geometrical 
drawings  ? 

The  objects  represented  in  geometrical  drawings  are  gen- 
erally solid  bodies,  with  irregular  or  curved  surfaces,  such 
as  houses,  blocks  of  wood,  chairs,  tables,  &c. 

3.  Can  we  generaUy^  conceive  of  their  shape  and  dimensions 
from  one  single  drawing  or  view? 

We  cannot.  For  instance,  if  we  place  ourselves  in  front 
of  a  house,  or  opposite  to  one  end  of  it,  or  if  we  stand 
behind  it,  or  look  down  upon  it  from  some  great  height, 
such  as  the  top  of  a  lofty  steeple,  we  shall  in  each  case 
have  a  different  view  of  it;  so  that,  unless  we  take  different 
drawings  of  it,  from  several  points,  it  will  not  be  possible 
to  convey  any  just  notion  of  its  general  appearance. 

4.  What  is  a  horizontal  plane? 

It  is  any  plane  parallel  to  the  water-level,  such  as  the 
level  ground,  the  floor  of  a  house,  &c. 


PRINCIPLES    OF    PLAN    DRAWING. 


71 


5;  What  is  a  vertical  plane  ? 

It  is  a  plane  perpendicular  to  a  horizontal  plane  ;  such 
as  the  front  or  ends  of  a  house,  or  the  face  of  a  vertical 
wall. 

6.  How  many  kinds  of  geometrical  drawings  are  necessary 
in  order  to  represent  the  form  and  dimensions  of  an  object  ? 

Three  kinds  only  are  necessary ;  viz.,  a  plan,  a  sec- 
tion, and  an  elevation. 

7.  What  is  a  plan  ? 

A  plan  of  an  object  merely  resembles  the  appearance 
which  it  would  present  to  the  eye,  when  viewed  from  a 
point  directly  .above  it.  , 

In  order  to  illustrate  this^  more  clearly,  let  us  proceed  to 
draw  the  plan  of  a  small  building 

In  commencing  a  building,  the  first  thing  necessary  is  to 
have  a  general  plan,  or  plan  of  the  foundation.  Let  us 
suppose  that  the  building  to  be  represented  is  a  cottage, 
with  a  door  and  window  only. 

First,  having  fixed  upon  the  scale 
on  which  the  drawing  is  to  be  made, 
say  30  feet  to  tlie  inch,  lay  off  the 
length  of  the  cottage  30  feet,  on  the 
line  ah,  and  the  width  24  feet,  on  dc. 
and  complete  the  rectangle  to  repre- 
sent the  exterior  dimensions  of  the  cottage  ;  that  is  to  say, 
its  length  and  breadth  from  out  to  out. 

Next,  lay  off  from  the  same  scale 
the  thickness  of  the  wall  from  a  to 
b,  and  from  a  to  c,  and  draw  the  in- 
terior rectangle,  having  its  sides  par- 
allel, respectively,  to  those  -of  the 
outer  one. 
wall. 


h  « 


This  will   rep?esent  the   interior  faces   of  the 


f2 


BOOK    III. SECTION    III. 


We  s^e  that  this  figure  has  nearly  the  same  appearance 
as  would  be  presented  by  the  foundations  of  a  small  build- 
ing, viewed  from  a  point  directly  over  them. 

Doors  and  windows  are  generally 
marked  in  a  ground  plan.  In  order 
to  distinguish  them  from  each  other, 
the  lines  of  the  foundation  walls, 
which  interfere  with  the   doors,  are 


rubbed  out.     The  doors  and  windows      I'        II 

will  be  marked  accordingly. 

The  complete  plan  of  the  cottage  is  now  drawn.  It 
shows  the  size  of  the  room,  the  thickness  of  the  walls,  and 
the  width  and  position  of  the  door  and  window. 

By  means  of  a  plan,  drawn  according  to  a  scale,  it  would 
be  easy  to  lay  out  correctly,  the  foundations  of  a  building 
and  the,  doors  and  windows  of  the  lower  story.  But  after 
building  a  few  courses,  we  should  be  obliged  to  stop  for 
want  of  further  directions,  because  the  plan  can  neither 
explctin  the  height  of  the  doors  or  windows,  nor  the  height 
of  any  other  part  of  the  building. 

This  proves  what  has  already  been  stated,  viz.,  that  more 
than  one  kind  of  drawing  of  any  object  is  always  necessary 
in  order  to  explain  its  form  and  dimensions.  Before  pro- 
ceeding to  the  other  kinds  of  geometrical  drawings,  men- 
tioned above,  we  will  add  some  further  explanations  and 
observations  on  the  subject  of  plans. 

8.  The  plan  of  any  object  is  always  supposed  to  be  made 
on  a  horizontal  plane  or  dead  level.  The  necessity  of  fol- 
lowing this  rule  will  appear  from  the  following  considera- 
tions. 

Suppose  it  were  required  to  build  a  house  on  uneven 
ground,  such,  for  example,  as  the  side  of  a  hill.  Every  one 
knows  that  in  laying  out  the  foundation,  no  reliance  would 
be  put  on  any  oblique  measurements  made  along  the  slope, 


PRINCIPLES    OF    PLAN    DRAWING.  73 

but  that  all  the  measurernents  would  have  to  be  made  in 
horizontal  lines.  For  instance,  if  you  were  to  measure  30 
feet  obliquely,  along  the  side  of  the  hill,  for  the  breadth  of 
your  proposed  building,  it  A^-ould  still  be  necessary  to  lay 
the  first  floor  horizontally.  After  this  was  done,  you  might 
find  the  space  which  was  laid  out  for  the  breadth  of  the 
building,  reduced  to  29  feet,  to  28  feet,  to  25  feet,  or  even 
to  a  less  distance,  according  to  the  steepness  of  the  slope 
of  the  hill.  The  plan  of  an  uneven  field,  in  which  the  di- 
mensions were  marked  according  to  oblique  measurements 
made  upon  the  sloping  or  irregular  surface  of  the  ground, 
would  therefore  be  of  no  use. 

9.  It  is  more  difficult  to  draw  the  plan  of  any  object 
having  sloping  or  oblique  lines,  than  to  draw  the  plan  of  a 
building  having  only  horizontal  and  vertical  lines,  because 
the  oblique  or  sloping  lines  must  all  be  reduced  in  a  certain 
proportion. 

10.  The  following  are  the  rules  for  laying  dovm  truly,  on 
a  horizontal  plane,  the  points  and  lines  of  all  objects,  any 
way  situated,  with  respect  to  it, 

11.  The  imaginary  horizontal  plane,  on  which  the  plan 
is  made,  and  to  which  all  points  and  lines  are  referred,  is 
called  the  horizontal  plane  of  projection. 

This  plane  may  be  so  taken  as  to  cut  the  object  which 
is  to  be  drawn  upon  it,  or  it  may  be  taken  directly  above  or 
below  the  object.  But  for  learners,  it  is  best  to  begin  by 
supposing  the  horizontal  plane  to  pass  through  the  base, 
or  lowest  point  of  the  given  object.  » 

In  respect  to  such  points  of  the  object  as  stand  upon  the 
plane  of  projection,  or  coincide  with  it,  there  can  be  no 
difficulty,  for  such  points  are  their  own  place  or  projections 
on  the  plane. 


74  BOOK    III. SECTION    III. 

From  every  point  without  the  plane  of  projection,  a  per- 
pendicular is  supposed  to  be  drawn  to  the  plane,  and  the 
point  in  which  this  perpendicular  pierces  the  plane,  will 
jmark  the  true  position  of  the  point  from  which  it  was 
drawn; 

If  the  plane  of  projection  be  supposed  tq  lie  below  the 
given  object,  then  all  the  points  of  the  object  will  be  above 
the  plane  of  projection ;  and,  consequently,  all  the  perpen- 
diculars, requisite  for  finding  the  position  of  the^e  points 
on  the  plane  of  projection,  will  go  downward  from  these 
points. 

But  if  the  plane  of  projection  be  supposed  to  be  above 
the  given  object,  then  the  several  points  of  the  object  will 
be  below  the  plane  ;  and,  consequently,  all  the  perpendicu- 
lars, necessary  for  finding  the  position  of  these  points  on 
the  plane  of  projection,  will  slope  up  from  the  given 
points. 

12.  Since  the  plane  of  projection,  in  plans,  is  always 
supposed  to  be  horizontal,  every  perpendicular,  whether 
dropped  or  raised,  will  be  a  vertical  or  plumb-line.  Con- 
sequently, if  we  suppose  two  plummets  to  be  suspended 
exactly  over  two  points  of  an  object,  the  plan  of  which  is 
required  to  be  drawn,  the  distance  between  the  plumb-lines, 
measured  perpendicularly,  will  be  the  true  distance  at  which 
the  two  points  ought  to  be  laid  down  on  the  plan. 

13.  To  explain  this,  draw  > 
three  lines  on  the  board  con- 
nected together ;  all  of  the 
same  length,  but  sloping  un- 
equally. These  may  repre- 
sent the  form  of  some  sloping 
or  oblique  object,  of  which  the 
plan  is  to  be  drawn.       \  '    - 


PRINCIPLES    OF    PLAN    DRAWING. 


75 


The  pupils  will  copy  this 
and  the  following  operations 
on  their  slates,  without  fur- 
ther directions,  until  the  figure 
is  completed.  From  the  ex- 
tremities of  each  of  the  three 
lines,  draw  dotted  lines,  paral- 
lel to  each  other,  directed  to- 
wards the  top  of  the  board. 

These  dotted  lines  may  rep- 
resent plumb-lines  held  over 
the  various  points  of  the  ob- 
lique object.  Now  mark  the 
various  points  of  the  oblique 
qbject  by  capital  letters  A,  B, 
C,  and  D,  from  left  to  right.  . 

As  the  distances  between 
the  four  plumb-lines,  repre- 
sented in  the  last  figure,  must 
be  measured  perpendicularly, 
not  obliquely,  draw  a  lin^e 
above  the  given  object,  and 
perpendicular  to  the  dotted 
lines,  on  which  the  said  distances  are  to  be  measured. 

At  the  points  where  the  perpendiculars  meet  the  horizontal 
line,  make  the  letters  a,  b,  c,  and  ^,  from  left  to  right. 

The.  distance  between  the 
points  a  and  b,  at  the  top  of 
the  figure,  represents  the  exact 
distance  between  the  plumb- 
lines  suspended  over  the  points 
A  and  B.  Consequently,  the 
perpendicular  line  ab,  at  the 
top  of  the   figure,  represents 


a 

b 

C 

tl 

i 

! 

-\ 

J 

"'.  _ 

i 

! 

i 
i    / 

/^ 

: 

^. ! 

j 

^ 

Nz, 

76 


BOOK    III. SECTION    III. 


a 

b 

z              d 

5^ 

C 

\ 

i 
i 

the  exact  length  which  ought  to  be  given  to  the  oblique 
line  AB,  in  drawing  a  plan  of  the  given  object. 

The  perpendicular  line  6c,  at  the  top  of  the  figure,  in  like 
manner,  and  for  the  same  reason,  represents .  the  exact  dis- 
tance which  ought  to  be  given  to  the  oblique  line  BC,  in 
the  plan  of  the  object. 

And  the  perpendicular  cdyZX  the  top  of  the  figure,  in  like 
manner  represents  the  exact  distance  which  ought  to  be 
given  to  the  oblique  line  CD^  in  the  plan. 

14.  Let  us  now  produce  the 
dotted  lines  below  the  given 
object,  and  draw  a  second  ho- 
rizontal line  intersecting  them 
perpendicularly;  and  let  us 
also  mark  the  points  of  inter- 
section by  the  same  letters 
«,  6,  c,  and  d.  C'        h  c  d 

Then,,  since  parallel  lines 
are  always  at  the  same  distance  from  each  other,  although 
produced  ever  so  far,  the  distance  between  the  points  «  and 
5,  at  the  bottom  of  the  figure,  will  be  equal  to  the  distance 
between  the  points  a  and  b  at  the  top ;  and  the  same  for  the 
distances  between  any  other  two  points. 

Consequently,  the  perpendicular  distances  ah,  he,  and  cd, 
at  the  bottom  of  the  figure,  will  be  equal  to  the  perpendicu- 
lar distances  ah,  be,  and  cd,  at  the  top ;  and,  therefore,  the 
lines  ah,  be,  and  cd,  at  the  bottom,  will  serve  equally  well 
to  represent  the  respective  lengths  which  ought  to  be  given 
to  the  oblique  lines  AB,  BC,  and  CD,  in  the  plan  of  the 
^iven  object. 

Hence  we  see,  that  either  the  upper  horizontal  line  ah^ 
or  the  lower  horizontal  line  ah,  may  represent  the  plane  ot 
projection,  ito  be  used  in  drawing  the  plan  of  the  oblique 
object ;  the  upper  line  will  represent  a  plane  passing  above 


PRINCIPLES    OF    PLAN    DRAWING.  77 

the  given  object,  and  the  lower  line  a  plane   passing  be- 
low it. 

This  illustrates  what  was  before  observed,  that  in  draw- 
ing the  plan -of  any  object,  it  makes  no  difference  whether 
the  plane  of  projection  is  taken  above  or  below  it. 

,15,  A  line  drawn  from  any  point  in  a  given  object,  and 
perpendicular  to  the  plane  of  projection,  is  called  the  pro- 
jecting line  of  the  pbint ;  and  the  place  where  the  perpen- 
dicular meets  the  plane,  is- called  the  projection  oi  the  point. 

16.  Let  us  illustrate  the  above  rules  by  means  of  a  square 
pyramid. — (Here  let  the  teacher  explain  the  shape  of  a 
square  pyramid,  and  exhibit  one  to  the  class.) 

If  we  look  down  upon  a  square  pyramid,  we  shall  see 
the  extremities  of  its  base,  its  vertex,  and  the  four  edges 
or  oblique  lines  which;  are  formed  by  the  meeting  of  its 
sides.  All  these  particulars  must  therefore  be  represented 
in  the  plan  of  a  square  pyramid. 

The  most  convenient  way,  is  to  suppose  the  horizontal 
plane  on  which  the  plan  is  to  be  made,  to  pass  through  the 
base  of  the  pyramid.  For  example,  if  we  place  the  pyra- 
mid upon  a  table,  the  level  surface  of  the  table  will  repre- 
sent the  plane  of  projection.  The  base  of  the  pyramid, 
now  standing  on  the  plane  of  projection,  coincides  with  it, 
and  will  be  its  own  projection,  without  any  enlargement  or 
diminution. 

The  base  of  the  pyramid  is  a 
square.  Represent  it,  therefore,  on 
the  paper  or  board,  by  drawing  a 
square  exactly  equal  to  it. 

In  the  present  instance,  the  base 
of  the  pyramid  coinciding  with  the 
plane  of  projection,  and  the  pyramid 
being  perfectly  regular,  it  is  evident 


78  BOOK    III. SECTION    III. 

that  a  perpendicular  dropped  from  the  vertex,  would  fall 
exactly  on  the  middle  or  central  point  of  the  base.  Mark, 
therefore,  the  middle  point  of  the  square,  and  it  will  repre- 
sent the  projection  of  the  vertex  of  the  pyramid. 

The  four  ridges,  or  oblique  lines,  remain  to  be  drawn. 
But,  one  extremity  of  each  of  these  lines  passes  through 
each  angular  point  of  the  base,  all  of  which  are  already 
marked  on  the  plan.  The  other  extremities  of  these  lines 
all  meet  at  the  vertex  of  the,  pyramid,  whose  projection  on 
the  plan  is  also  determined.  There- 
fore, draw  from  the  centre  of  the 
base  four  straight  lines,  one  to  each 
angle  of  the  base,  and  they  will 
represent,  in  the  plan,  the;  four  ridges 
of  the  pyramid. 

You  see  that  the  plan  of  the  pyra- 
mid, now  drawn  upon  paper,  shows 
no  dimensions  but  those  of  the  base. 

It  also  indicates  the  particular  point  of  the  base  over  which 
the  vertex  stands ;  but  it  neither  explains  the  height  of  the 
pyramid,  nor  the  obliquity  or  slopes  of  its  sides. 

The  plan,  therefore,  cannot  alone  explain  the  nature 
either  of  a  building  or  pyramid,  or  of  any  other  object,  and 
recourse  must  be  had  to  some  other  kind  of  drawings.  > 

OF    SECTIONS. 

17.  A  section  is  a  plane  figure^  formed  by  cutting  any 
solid  body  into  two  parts.  A  solid  body  may  be  cut  in  a 
great  number  of  directions  :  viz.,  horizontally,  vertically,  and 
obliquely:  and  hence,  the  number  of  sections  which  may 
be  formed  of  any  object,  are  infinite,  or  beyond  calcula- 
tion. 

To  avoid  the  confusion  which  might  arise  in  plan-draw- 
ing, from  sections  taken  at  random,  the  geometrical  drawing 


OF    SECTIONS.  79 

called  a  spction,*ts  always  taken  vertically ;  that  is  to  say, 
the  object  is  supposed  to  be  cut  right  down,  perp'endicularly, 
from  top  to  bottom,  by  a  vertical  plane  ;  in  other  words,  it 
is  supposed  to  be  cut  everywhere  in  a  plumb-line. 

A  section  is  principally  intended  to  show  the  heights  of 
objects,  and  thereby  to  make  up  for  the  defects  of  the  plan, 
which  have,alr0ady  been  explained. 

Supposing  it  were  required  to  measure  the  height  of  one 
of  the  sides  of  a  room.  This  could  not  be  correctly  done 
by  measuring  diagonally  or  obliqfuely — that  would  be  quite 
wrong.  There  is  no  way  of  finding  the  true  height  except 
by  measuring  vertically,  or  in  the  plumb-line. 

If,  then,  we  suppose  a  section  of  the  room  to  be  taken 
in  which  we  now  are,  it  is  evident  that  if  the  section  were 
taken  in  a  sloping  direction,  it  would  cut  the  sides  of  the 
room  obliquely.  Such  a  section  would  therefore  give  an 
erroneous  representation  of  the  sides  of  the  room. 

Sections  taken  across  any  building  or  object,  will  of 
course  serve  to  show  the  breadth  as  well  as  the  height  of 
it9  varions  parts.  In  order'  that  this  may  be  done  truly, 
another  rule  must  be  laid  down  no  less  essential  than  the 
former:  viz., 

In  taking  the  section  of  any  regular  object,  such  as  a 
rectangular  building,  the  object  is  always  supposed  to  be 
cut  right  across;  that  is,  in  a  direction  perpendicular  to 
two  opposite  sides ;  and  the  same  reason  holds  good  in  this 
case,  which  was  given  for  employing  a  vertical  plane. 

Supposing-  we  wished  to  measure  the  breadth  of  this 
room.  You  see  at  once,  that  if  we  took  the  measurement 
obliquely,  from  angle  to  angle,  the  result  would  be  quite 
wrong ;  and  that  there  is  no  possible  way  of  measuring  the 
breadth  of  the  room  accurately,  except  in  a  direction  per- 
pendicular to.  its  two  opposite  sides. 

From  these  considerations  it  must  be  evident,  that  any 


80 


BOOK    III. SECTION    III. 


section  of  a  building,  or  of  an  object;  taken  in  a  sloping  or 
oblique  direction,  would  not  be  of  the  smallest  use,  because 
it  would  either  misrepresent  the  height,  or.  the  breadth,  or 
both. 


d 

:C 


18.  This  being. premised,  let  us  now  proceed  to  draw 
a  section  of  the  small  cottage,  of  which  we  have  already 
drawn  the  plan. 

Let  us  suppose  that  the  proposed  section  is  required  to 
pass  through  the  door  of  the  building.  Draw  a  dotted  line 
perpendicularly  across  the  plan 
of  the  cottage,  passing  through 
the  door.  This  dotted  line  will 
represent  the  direction  in  which 
the  proposed  section  is  to  be 
taken.  « 

Mark  the  points  on  the  plan  where  the  dotted  line  cuts  the 
front  and  back  walls  of  the  cottage,  by  the  letters  a,  b,  c, 
and  d.  The  distances  between  the  points  a,  b,  c,  and  d,  show 
the  breadth  of  the  cottage  and  the  thickness  of  the  walls. 

As  the  same  dimensions  which  have  been  used  in  the 
plan  must  be  again  represented  in  the  section,  it  will  save 
time  to  transfer  the  whole  of  them,  at  once,  from  the  plan 
to  the  section. 

Therefore,  draw  a  separate  line  to  represent  the  level 
of  the  building,  which  will  also  be  the  ground  line  or  base 

d 


c  d 


of  the  section.     Then  divide  this  line  in  the  same  manner 
as  the  dotted  line  abed  is  divided  in  the  plan. 


PRINCIPLES    OF    PLAN    DRAWING. 


81 


Under,  the  respective  points  of  division,  on  this  new  line, 
mark  the  same  letters  a,  h,  c,  and  d.  When  this  is  done, 
the  corresponding  or  like  parts  of  both  lines  M^ill  be  knovi^n 
by  inspection.  . 

From  the  points  a,  b,  c,  and  c?,  on  the 
ground  line  of  the  section,  which  rep- 
resents the  position  and  thickness  of  the 
walls  of  the  cottage,  raise  perpendiculars 
to  show  the  height  of  the  walls.  Join 
the  tops  of  these  perpendiculars  by  a  dotted  line  which  will 
be  horizontal,  and  this  line  will  show  the  level  from  whence 
the  roof  is  supposed  to  spring. 

The  ^plan  of  the  cottage  is  still  supposed  to  remain  "on  the 
hoard  and  slates,  but  is  left  out  in  some  of  the  following 
figures.  It  will  again  be  occasionally  introduced,  whenever 
it  shall  be  necessary  to  point  out  the  connection  between  the 
plan  and  the  section,       •  '        ' 


19.  We  will  now  suppose  the  roof  to  be 
a  regular  pitch  roof.  Therefore,  bisect 
the  last-drawn  line,  in  order  to  find  the 
middle  of  the  building ;  and  from  the  point 
of  bisection  raise  a  perpendicular,  to  show 
the  height  of  the  roof.  From  the  extremi- 
ties of  this  perpendicular,  draw  an  oblique 
line  to  the  outside  of  the  top  of  each  wall :    ^  b  .    c  d 

this  will  show  the  sides  of  the  roof.  Then  draw  right  lines 
interiorly,  parallel  to  the  last  lines,  to  show  the  thickness 
of  the  roof. 

As  the  section  is  supposed  to  pass  through  the  door  of 
the  cottage,  a  line  must  be  drawn  to  represent  the  top  of 
the  door,  and  tO  show  the  height  of  it.  ^ 

The  section  which  has  just  been  drawn,  is  only  intended* 

to  give  a  general  notion  of  this  kind  of  geometrical  drawing. 

4* 


82 


BOOK    III. — SECTION    III. 


Many  particulars  are  therefore  omitted,  which  it  would  be 
proper  to  introduce  into  a  finished  section  of  a  building. 
For  instance,  the  depth  and  thickness  of  the  foundation, 
the  recess  of  the  door,  the  thickness  of  the  rafters  and  other 
parts  of  the  roof;  also,  its  projection  over  the  walls,  if 
formed  with  eaves.  These,  and  other  details  might  easily 
have  been  represented,  by  adding  a  few  more  lines.     The 


d 

\c 

Plan. 

1    1        M 

ah  c  d 

rough  section  of  the  cottage  is  now  complete,  and  you  may 
observe,  that  those  dimensions  which  are  marked  with  the 
same  letters,  agree  in  both. 

The  plan  and  sections,  as  they  stand  at  present,  explain 
sufficiently  the  general  dimensions  of  the  cottage,  and  the 
proportions  of  the  roof  and  door;,  but  they  do  not  show  the 
height  of  the  windows,  nor  the  general  appearance  of  the 
building. 

The  latter  particulars  cannot  be  represented  without  the 
assistance  of  the  third  kind  of  geometrical  drawing,  before 
mentioned,  called  an  elevation. 


OF    THE    ELEVATION. 

20.  An  ELEVATION  is  the  view  of  any  upright  side  of  a 
building  or  other  object,  nearly  such  as  it  "would  appear 
to  a  person  standing  directly  in  front  of  it. 

In  order  to  understand  this  definition  more  clearly,  let 
us  draw  an  elevation  of  the  front  of  the  cottage. 


OF    THE    ELEVATION. 


83 


— 

I. 

n 

i 
j 

1- 

1 

r 

i      1      «        1       1 

2    3 


4    5 


•As  the  principal  dimensions  of 
the  front  of  the  eottage  appear  in 
the  plan,  let  the  various  points  be 
marked  by  the  figures  1,  2,  3,  4,  5, 
and  6. 

The  points  thus  marked  show 
the  length"  of  the  front  of  the  cot- 
tage,  and  the  breadth  and  position  of  the  door  and  win- 
dow. 

As  all  these  dimensions  must  appear  in  the  elevation  of 
the  cottage,  the  easiest  method  will  be  to  transfer  them 
from  the  plan  to  the  elevation  at  once. 

Draw,  therefore,  a  separate  line,  to  represent  the  ground 
line,  or  level  upon  which  the  front  of  the  cottage  stands  ;  and 
upon  this  line,  set  off  a  distance  equal  to  the  length  of  the 
cottage,  and  divide  it  in  the  same  manner  as  the  front  of 
the  cottage  is.  divided  in  the  plan. 

Mark  also  the  various  points  of  division  on  this  new  line, 
by  the  figures  1,  2,  3,  4,  5,  and  6.     When  this  is  done,  the 


2  3 


4    5 


2  3 


4  5 


corresponding  or  equal  parts  in  the  plan  and  in  the  ground 
line  of  the  elevation,  are  known  by  inspection.  From  the 
points  1  and  6  of  the  ground  line  of  the  elevation,  let  per- 
pendiculars be  drawn  to  show  the  height  of  the  walls.  Now, 
since  the  height  of  the  walls  is  already  represented  in  the 
settion,  take  that  height  in.  the  dividers  and  lay  it  off  on 
the  perpendiculars  through  1   and  6.  .     .  • 


84 


BOOK    III.^-SECTION    III. 


Join  the  top  of  these  perpendiculars  by  a  straight  line. 
This  line  will  represent  the  bottom  of  the  roof  of  the  cottage. 


1       23         45       6  123         45       6 

From  the  points  2  aid  3^  of  the  ground  line  of  the  eleva- 
tion, which  represent  the  width  of  the  door,  raise  perpen- 
diculars to  show  the  height  of  the  door.  Find  the  proper 
length  of  these  perpendiculars  by  measuring  the  height  of 


123         456  123         456 

the  door  in  the  section,  and  then  transfer  it  to  the  elevation. 
Complete  the  form  of  the  door  by  joining  the  top  of  the 
above  perpendiculars. 

From  the  points  4  and  5  in  the  elevation,  which  mark 
the  position  of  the  window,  raise  perpendiculars  to  find  the 
sides  of  the  window.     Next  complete  the  window  by  draw- 


nr 


2    3 


4    5       6 


ing  the  top  and  bottom  of  it,  ai  their  proper  height.  Dot 
that  part  of  each  of  the  last  perpendiculars,  which  falls 
below  the  bottoms  of  your  windows. 


i   UNIVERSITY  ) 

OF    THE    ELEVA™N  ^''[.       J^  S5 

The  form  of  the  roof  is  now  alone  wanting.  The  length 
of  the  roof  must  of  course  be  equal  to  the  length  of  the 
building,  and  the  height  of  it  may  be  found  by  referring  to 
the  section.  ■ 

It  is  a  general  rule,  in  geometrical  elevations,  never  to 
represent  the  height  of  any  sloping  object  by  oblique  meas- 
urements taken  along  the  slope  ;  but,  by  dropping  a  perpen- 
dicular from  the  highest  point,  or  vertex  of  the  slope,  to  the 
level  of  the  lowest  point  or  base  of  it. 

In  short,  the  height  of  any  sloping  object  in  a  geometrical 
elevation  is  measured  by  that  perpendicular  line,  which 
would  be  called  the  altitude  of  any  similar  figure  or  body, 
in  Geometry.  Therefore,  in  transferring  the  height  of  the 
roof  from  the  section  to  the  elevation,  make  it  in  the  eleva- 
tion equal  to  the  dotted  perpendicular  drawn  in  the  section. 
Next  draw  the  roofs :  wh^n  this  is  done,  the  drawings  of 
the  cottage  are  as  follows : 


Plan. 


Section. 


Elevation. 


21.  The  Plan,  Section,  and  Elevation  of  a  small  cottage 
are  now  complete,  and  from  these  three  geometrical  draw- 
ings put  together,  every  dimension  necessary  for  explaining 
the  proportions  of  the  building  may  be  known. 

The  length  of  the  building  is  shown  in  the  plan  and  ele- 
vation, and  is  the  same  in  both. 

The  breadth  of  the  building,  and  thickness  of  the  walls, 


86  BOOK    III. SECTION    III. 

are  shown  in  the  plan  and  section,  and  are  the  same  in 
both. 

The  breadth  of  the  doqr,  and  that  of  the  window,  are 
shown  in  the  plan  and  elevation. 

The  height  of  the  door  is  shown  in  the  gection  and  ele- 
vation, and  is  the  same  in  both. 

The  height  of  the  window  is  shown  in  the  elevation  only. 
But  if  the  section  had  been  taken  through  the  window,  in- 
stead of  the  door,  then  the  height  of  the  window  would 
have  been  shown,  and  not  that  of  the  door. 

The  height  of  the  walls,  arid  the  perpendiculgir  height 
of  the  roof,  are  shown  in  the  section  and  elevation,  and 
are  equal  in  both.  But  the  particular  form  of  the  roof  is. 
clearly  explained  in  the  section  only. 

REMARKS    ON    ELEVATIONS. 

22.  An  elevation  is  always  supposed  to  be  drawn  on  a 
vertical  plane,  which  is  called  the  vertical  plane  of  projeo 
tion. 

Those  points  of  an  object  which  lie  in,  or  coincide  with, 
the  plane  on  which,  the  elevation  is  drawn,  are  their  own 
projections  on  that  plane.  Those  points  of  the  given  object 
which  lie  without  the  plane  of  projection,  must  be  trans- 
ferred to  it,  bylines  drawn,  from  the  points  and  perpendicular 
to  the  plane  of  projection.  Such  lines  are  called  projecting 
lines. 

Since  all  the  projecting  lines  which  determine  an  eleva- 
tion are  perpendicular  to  a  vertical  plane,  they  must  neces- 
sarily be  horizontal.  The  walls  or  sides  of  a  building  are 
vertical  planes,  being  built  according  to  a  plumb-line  ;  and 
tlierefore,  in  taking  a  geometrical  elevation,  the  plane  on 
which  it  is  made  may  be  supposed  to"  coincide  with  the  front 
of  the  building,  or  any  other  side  which  is  to  be  repre- 
sented. 


REMARKS    oN    ELEVATIONS.  87 

When  this  is  done,  the  length  and  height  of  the  side  of 
the  building,  and  the  height  and  breadth  of  the  doors  and 
windows,  &c.,  may  be  laid  down  in  a  geornetrical  elevation, 
according  to  their  actual  dimensions  from  measurement. 

The  roof,  from  its  sloping  figiire,  is  the  only  part  of  the 
exterior  side  of  the  building  which  cannot  agree  with  the 
plane  of  projection ;  and  hence,  in  drawing  the  elevation 
of  the  cottage,  it  was  necessary  to  diminish  the  oblique 
lines  of  the  slope  of  the  roof,  in  order  to  find  the  true  ver- 
tical height  of  it.  They  were  diminished  in  the  same  way 
that  the  oblique  lines  are  diminished  in  a  plan,  in  order  to 
find  the  base  of  any  slope. 

23.  It  is  not  necessary,  in  a  geometrical  elevation,  that 
the  plane  of  projection  should  be  supposed  to  agree  ex- 
actly with  the  upright  side  of  the  building  or  object  which 
is  to  be  represented.  But  when  they  do  not  agree,  it  is 
necessary  that  the  plane  of  projection  should  be  parallel 
to  the  upright  side  of  the  building  or  object,  of  which  the 
elevation  is  to  be  drawn.  In  that  case,  the  projecting  per- 
pendicular will  form,  on  the  plane  of  projection,  a  figure 
exactly  similar  to  the  front  of  the  building  or  object.     Con- 

vsequehtly,  if  you  suppose  a  plane  of  projection  to  be  chosen, 
parallel  to  the  upright  side  of  a  building  or  other  object,  of 
which  an  elevation  is  required,  then' the  dimensions  of  the 
various  parts  of  the  upright  side  of  the  given  object  may 
be  laid  down  in  the  drawing  in  their  true  proportions,  ac- 
cording to  measurement. 

24.  From  the  figures  which  have  been  drawn,  and  the 
instructions  which  have  been  given,  on  the  subject  of  Plan- 
drawing,  it  appears  that  plans  and  elevations  are  drawn 
ex&ctly  according  to  the  same  principles,  with  only  this 
difference  :  that  in  a  plan,  the  plane  of  projection  is  always 


88  BOOK    III.— SECTION    III. 

horizontal,  whereas,  in  an  elevation,  it  is  always  verti- 
cal. -' 

All  horizontal  planes,  which  may  be  used  as  planes  of 
projection  for  drawing  the  plan  of  a  building,  will  be  parallel 
to  each  other;  but  the  vertical  planes,  on  which  the  ele- 
vations are  drawn,  may  be  oblique,  perpendicular,  or  paral- 
lel to  each  other. 

For  example,  the  several  floors  of  any  building,  being  all 
level,  and  all  the  points  of  each  at  the  same  height  from 
the  ground,  are  horizontal  planes- parallel  to  each  other. 
But  of  the  walls  of  a  building,  which  are  all  vertical  planes, 
some  two  of  them  may  be  perpendicular  to  each  other,  such 
as  the  side  and  end  walls ;  while  others  may  be  oblique  to 
each  other,  as  is  often  seen  in  irregular  buildings. 

OF    OBLIQUE    ELEVATIONS. 

25.  If,  in  drawing  the  elevation  of  any  rectangular  build- 
ing, the  plane  of  pi-ojection  were  chosen  oblique  to  one  of 
the  sides,  instead  of  parallel  to  it ;  then,  the  length  of  that 
side  of  the  building  and  the  breadth  of  the  doors  and  win- 
dows would  be  diminished  in  the  drawing,  in  such  a  manner 
as  to  give  a  false  notion  of  the  object.  In  an  oblique  eleva- 
tion of  this  kind,  the  projecting  lines  which  are  drawn  per-4 
pendicular  to  the  pUne  of  projection,  will  be  oblique  to  the 
building ;  and  hence,  all  the  dimensions  except  those  which 
are  vertical  would  be  diminished  or  misrepresented  in  the 
drawing:  hence,  such  elevations  are  of  little  use,  and  are 
therefore  seldom  made. 

But,  although  oblique  elevations  of  the  fronts  of  buildings 
are  seldom  made,  it  often  happens  that  the  front  of  a  fine 
building  is  ornamented  with  columns,  mouldings,  and  archi- 
tectural decorations,  many  parts  of  which  are  oblique  to  the 
general  plane  of  the  front  of  the  building,  beyoiid  which 
they  project. 


OF    OBLIQUE    ELEVATIONS. 


89 


The  proper  methods  of  representing  such  Ornaments  in 
geometrical  elevations,  cannot  therefore  be  well  understood, 
unless  the  principle,  according  to  which  oblique  elevations 
of  any  upright  object  may  be  drawn,  is  clearly  explained. 

This  being  premised,  we  shall  give  an  example  of  tiie 
method  of  di-awing  an  oblique  elevation  of  the  Cottage,  of 
which  we  have  already  drawn  the  plan,  and  section,  and 
geometrical  elevation. 


26.  Resume  the  plan*  before  drawn,  and  mark  thereon 
the  points  1,  2,  3,  4,  5,  and  6. 

A    straight    line    must   next   be 
drawn,  to  represent  the  new  plane 
of  projection,  on  which  the  oblique 
elevation  is  to  be  made.    This  new 
plane  of  projection  may  either  be       12   3        4    5      6 
supposed  to   coincide  with   some  line  of  the  front  face  of 
the  building  or  not.     We  shall  take  it  to  coincide  or  agree 
with  that  extremity  of  the  front  of  the  building  which  is 
marked  by  the  figure  6. 

Draw,  therefore,  a  right  line  through  the  point  6,  forming 
an  acute  angle  with  the  front  of  the  building,  and  this  line 
will  represent  the  new  plane  of  projection,  which  is  vertical. 

From  the  various  points  of 
the  front  of  the  building,  draw 
perpendiculars  to  the  last  line, 
and  dot  them.  These  perpen- 
diculars will  determine  the  true 

places  of  tRe  points  in  the  ob-     i \%  k^ 

lique  elevation.  \       \    \ 

Mark,  therefore,  in  like  man-      ^--'''^^r^ 
ner,  by  the  figures  1,  2,  3,  4,  5,  "•  . 

and  6,  the  several  corresponding  points  on  that  line  which 
represents  the  plane  of  projection. 


« 


90  BOOK    III, SECTION    III. 

That  end  of  the  cottage  which  is  nearest  to  the  plane  of 
projection  must  also  be  represented.  One  extremity  of  it 
coincides  with  the  said  plane.  From  the  other  extremity 
draw  a  dotted  perpendicular  to  the  plane  of  projection,  and 
mark  the  corresponding  points,  at  the  ends  of  this  line,  by 
/he  figure  7. 

The  distance  between  the  points  1  and  6  in  the  plan, 
shows  the  length  of  the  front  of  the  cottage  ;  and  there- 
fore the  distance  between  the  corresponding  points  1  and  6, 
on  the  plane  of  projection,  will  also  represent  the  length 
which  ought  to  be  given  to  the  front  of  the  cottage  in  the 
oblique  elevation. 

The  distance  between  the  points  6  and  7,  in  the  plan, 
represents  one  end  of  the  cottage;  and  therefore  the  dis- 
tance between  the  corresponding  points  6  and  7,  on  the 
plane  of  projection,  will  also  represent  the  length  whidi 
ought  to  be  given  to  that  end  of  the  cottage  in  the  oblique 
elevation. 

And,  in  like  manner,  as  the  breadths  of  the  door  and  win- 
dow are  represented,  respectively,  by  a  certain  distance  in 
the  plan  ;  so  the  same  dimensions,  in  the  oblique  elevation, 
must  be  represented  by  the  distance  between  the  corre- 
sponding points,  on  the  plane  of  projection. 


I      1     I ' — M — t- 


\  12   3       4567 

You  will,  therefore,  draw  a  line  for  the  ground  line  of 
the  oblique  elevation.  Divide  this  line  in  the  same  manner 
as  the  one  which  represents  the  plane  of  projection,   and 


OF    OBLIQUE    ELEVATIONS. 


91 


2     3        4     5 


mark  it  with  the  same  numeral  figures.  From  the  points 
1,6,  and  7,  on  the  ground  line 
of  the  elevation,  raise  perpen- 
diculars equal  to  the  height  of 
the  cottage  ;  and  draw  the  up- 
per-line.  At  2  and  3  also 
draw  perpendiculars,  and  lay 
off  the  height  of  the  door ;  and 
do  the  same  at  4  and  5  for  the 

window — dotting  those  parts  of  the  perpendiculars  which  lie 
below  the  window.  ^ 

The  roof  only  remains  to  be  drawn.  Before  this  can  be 
done,  it  will  be  necessary  to  find  the  points  where  the  ridge 
ought  to  be  laid  down  in  the  plane  of  projection. 

The  ground  plan  of  the  cottage  does  not  show  the  ridge 
of  the  roof;  but  it  is  evident  that  the  ridge  of  a  regular  roof 
with  a  simple  pitch,  must  be  directly  over  the  middle  of  the 
building. 

In  OTder  to  save  the  trouble 
of    drawing   a    separate   plan, 
draw  a  dotted  line  RR  through  R 
the  middle  of  the  plan  already 
drawn,  to  represent  the  ridge. 

From  the  points  R  and  R, 
which  represent  the  extremities 
of  the  ridge  of  the  roof,  draw 
dotted  perpendiculars  to  the 
plane  of  projection,  and  mark  j^ 
the  points  where  they  meet  the 
plane,  by  the  letters  R  and  R. 
The  points  R  and  R  must  next 
be  transferred  to  the  ground 
line  of  the  oblique  elevation. 
From  these  new  points  R  and  R,  draw  perpendiculars  and 


92 


BOOK    III. — SECTION    III. 


lay  off  the  height  of  the  cottage,  which  is  found  by  referring 
to  the  section :  then  draw  the  upper  line,  which  will  repre- 
sent the  ridge ;  after  which,  draw  oblique  lines  from  the 
extremities  of  the  ridge  to  the  proper  points,  in  order  to 
complete  the  form  of  the  roof. 

The  oblique  elevation  of  the  cottage  is  now  finished,  as 
below,  where  the  parallel  elevation  is  also  given. 


1  ie2  3  4   5       6  R  7 

The  heights  of  the  various  parts  of  the  oblique  elevation 
agree  with  those  of  the  section  and  front  elevation ;  but  all 
the  other  dimensions  are  changed,  being  less  than  they  were 
in  the  plan! 

If  the  plane  on  which  the  oblique  elevation  was  made 
had  formed  a  greater  angle  with  the  front  of  the  building, 
then  the  various  dimensions,  in  that  part  of  the  oblique  ele- 
vation which  represents  the  front  of  the  cottage,  would  have 
been  still  more  diminished. 

In  the  direct  elevation  of  the  front  of  the  cottage,  it  was 
not  necessary  to  take  any  notice  of  the  points  R  and  R, 
because  they  fell  directly  over  the  ends  of  the  building. 
The  two  ends  of  the  building  being  perpendicular  to  the 
plane  of  projection,  will  fall  in  the  parts  of  the  vertical 
lines  through  R  and  R,  which  lie  between  the  ridge  RR 
and  the  upper  line  of  the  front. 

27.  In  transferring  the  several  heights  from  the  section 
to  the  elevations,  each  dimension  was  measured  separately, 


OF    OBLIQUE    ELEVATIONS. 


93 


R 


tXX 

"" "d  " 


one  after  the  other ;  but  it  is  best  to  transfer  the  various 
heights  from  a  section  to  an  elevation  all  at  once,  in  the 
same  manner  as  the  dimensions  are  transferred .  from  the 
plan  to  the  elevation. 

Remember  that  the  section  of  the  cottage  was  formed  by 
a  plane  cutting  it  through  the 
door,  and  perpendicular  to  the 
front,  and  that  ar  is  the  line 
in  which  such  plane  cuts  the 
front.  At  a  convenient  dis- 
tance from  the  section  draw 
the  vertical  line  AR.  Then, 
through  the  various  points  of 
the  section  whose  heights  you 

wish  to  note,  draw  the  dotted  horizontal  lines  RR,  rr,  dd, 
and  Aa,  and  note  the  points  in  which  they  cut  the  vertical 
line  RA. 

The  distance  between  the  two  points  a  and  d  in  the  sec- 
tion, represents  the  height  of  the  door  in  the  cottage ;  and 
the  distance  between  the  two  corresponding  points  A  and  d^ 
on  the  vertical  line  AR,  will  represent  the  height  which 
ought  to  be  given  to  the  door  in  a  geometrical  elevation; 
and  the  same  for  all  other  points.  Hence,  all  the  points 
necessary  for  transferring  the  several  heights  of  the  front 
of  the  building  from  a  section  to  an  elevation,  are  now 
marked  on  the  vertical  line  AR. 


28.  If  you  wish 'to  draw  the  front  elevation  from  the  sec- 
tion and  plan,  draw  a  line  to  represent  the  ground  line. 
Then  draw  a  vertical  line  AR,  and  make  it  equal  to  ar, 
which  shows  the  height  of  the  cottage  ;  and  lay  off  in  the 
same  manner  the  height  of  the  door  ad,  and  the  height  of  the 
wall  ar.  Then,  at  a  convenient  distanae  from  a,  mark  the 
comer  of  the  cottage  1  ;  and  from  the  plan  lay  off  the  dis- 


94 


BOOK    III. SECTION    III. 


tances  from  1  to  2,  1  to  3,  4,  5,  and  6  ;  and  through-  2,  3, 

R 





A  a  1       2    3         4     5       6 

and  6  draw  perpendiculars,  which  being  met  by  parallels 
through  d,  r,  and  R,  will  determine  all  the  parts  of  the  cot- 
tage. The  two  heights  aw,  aw,  to  the  bottom  and  top  of  the 
window,  are  not  found  in  the  section,  but  are  taken  from  the 
oblique  elevation. 

29.  After  finishing  an  elevation,  or  other  geometrical 
drawing,  the  superfluous  or  dotted  lines  representing  planes 
of  projection,  scales  of  heights,  &c.,  are  rubbed  out ;  except- 
ing only  those  imaginary  lines,  marked  in  the  plan,  which 
show  the  direction  according  to  which  the  sections  or  ob- 
lique elevations  accompanying  the  plan  may  have  been 
taken. 

Let  us,  therefore,  rub  out  the  superfluous  lines,  letters, 
&c.,  in  the  figures  which  have  been  drawn,  leaving  only 
such  as  are  necessary  to  explain  the  connection  between 
the  plan  and  section,  and  between  the  plan  and  oblique 
elevation.     We  shall  then  have^ 


•i  1-      II 

Plm. 


Section. 


GENERAL    REMARKS. 


95 


Front  Elevation.  Oblique  Elevation. 

.  <  The  Plan,  Section,  and  Elevation,  completed  above,  are 
sufficient  to  give  a  full  insight  into  the  principles  of  plan 
drawing. 


GENRRAL    REMARKS. 


30.  AH  plans,  sections,  and  elevations  are  drawn  by  lay- 
ing down  a  certain  number  of  points  and  lines  truly,  on 
some  plane  surface,  according  to  geometrical  principles.  In 
drawing  some  objects,  it  maybe  necessary  to  lay  down  a 
great  number  of  points  and  lines:  in  others,  only  a  few;  but 
whether  the  number  be  great  or  small,  each  individual  point 
or  line  must  be  drawn,  in  all  cases,  according  to  some  one 
or  other  of  the  foregoing  rules. 

Plans  which,  as  before  stated,  resemble  the  appearance 
of  any  object  viewed  from  a  height  directly  above  it,  do  not 
carry  a  very  just  notion  of  tjie^  object  to  persons  ignorant 
of  the  principles  of  plan  drawing ;  because  opportunities  of 
looking  perpendicularly  or  directly  down  upon  objects  are 
not  common. 

Ground  plans,  or  foundation  plans  of  buildings  or  other 
works,  do  not  give  any  just  notion  of  the  appearance  of  the 
object  represented;  because  when  a  building  is  finished,  it 
is  impossible,  from  any  point  of  view  whatever,  to  see  the 
various  walls  and  foundations,  in  the  manner  in,  which 
they  must  be  represented  in  the  ground  plan,  the  whole  of 
these  parts  being  hidden  by  the  roof.     In  fact,  the  ground 


96  BOOK    III* — -SECTION    III. 

plan  of  any  finished  building  is,  properly  speaking,  a  hori- 
zontal section  through  the  various  walls — the  only  difference 
between  it  and  the  common  section  consisting  in  this,  that 
the  common  section  is  taken  vertically,  whereas  the  sec- 
tion which  exhibits  the  ground  plan  is  taken  horizontally. 

31.  In  plans,  sections,  and  elevations  of  any  object,  when 
the  various  points  and  lines  have  been  laid  down  according 
to  the  rules  of  projection,  it  is  usual  afterwards  to  color  or 
shade  the  figure  in  order  to  make  a  finished  drawing. 

The  art  of  plan  drawing,  therefore,  comprehends  two  dis- 
tinct operations  :  first,  the  projection  of  the  lines  which  form 
the.  representation  of  the  object;  and  secondly,  the  shading 
or.  coloring  of  it. 

In  colored  plans  and  sections,  masonry  is  generally  made 
red;  wood  so  as  to  represent . its  own  natural  color;  earth 
of  a  sandy  color ;  iron  of  a  dark  blue  ;  and  water  of  a  light- 
ish blue. 

In  plans  not  colored,  masonry  is  generally  made  dark, 
while  wood  and  other  substances  are  made  lighter. 

In  sections  not  colored,  different  substances  are  shaded 
darker  or  lighter,  according  to  the  fancy  of  the  draughtsman. 

In  plans  of  buildings,  the  doors  and  windows  are  left 
blank,  while  the  walls  are  either  colored  or  shaded.  And 
in  sections,  a  marked  distinction  of  color  or  shade  is  also 
made  between  the  solid  part  of  the  walls,  and  the  doors, 
windows,  or  other  apertures  which  may  be  representfed. 

In  elevations  of  any  object,  whether  colored  or  not,  the 
various  parts  are  shaded  in  such  a  manner  as  to  resemble, 
as  much  as  possible,  the  outward  appearance  of  the  object. 


OF    ARCHITECTURE.  97 


BOOK  IV. 


SECTION  r. 

OF    ARCHITECTURE. 

1.  What  is  Architecture? 
Architecture  is  the  art  of  construction. 

2.  Into  how  many  branches  is  it  divided  ? 
Into  three  principal  parts : — 

1st.  Civil  architecture,  which  embraces  the  construction 
of  public  and  private  edifices. 

2d,  Naval  urchitecture,  which  embrace€^  the  construction 
of  vessels,  ports,  artificial  harbors,  &;c. ;  and 

3d.  Military  architecture,  which  embraces  the  construc- 
tion of  forts,  redoubts,  and  all  military  defences.  We  shall 
speak  here  only  of  civil  architecture. 

3.  What  are  the  elements  of  architecture  ? 
They  are  the  mouldings. 

4.  What  are  mouldings  1 

They  are  the  projecting  parts  which  serve  to  ornament 
architecture. 

5.  How  many  kinds  of  mouldings  are  there  ? 

-  Three  kinds :  those  bounded  by  planes ;  those  bounded 
by  curved  surfaces ;  and  those  bounded  by  both  plane  and 
curyed  surfaces.  ^ 


98  BOOK    IV. SECTION    I. 

6.  What  are  the  principal  plane  mouldings  ? 

They  are  the  Filletj  the  Drip,  and  the  Plate-band. 

7.  What  is  a  fillet  1 

It  is  a  square  moulding  which  projects  over  a  distance 
equal  to  its  height. 

8.  What  is  a  drip  ? 

It  is  a  large  projecting  moulding,  hollowed  on  the  under 
side,  and  placed  in  cornices  to  protect  the  edifice  from  rain 

9.  What  is  a  plate-hand  1 

It  is  a  large  and  flat  mou^ding  which  projects  but  little. 

10.  What  are  the  principal  circular  mouldings  ? 

The  Ovolo,  the  bead  or  Astragal,  the  Torus,  the  Cavetto, 
the  Scotia,  the  Cyma -recta,  the  Cyma-reversa,  and  the  Ogee. 

11.  What  is  an   Ovolo,  and  how  do  you  trace  it  ? 

An  ovolo  is  a  moulding  flat  on  the  top  and  bottom,  and 
whose  circular  projection  is  equal  to  its  height. 

To  describe  it,  make  the  perpendicular  height  AD  equal 
to  the  projection  AC:  then,  with^  as  a  centre,  describe 
the  arc  DC.  If  you  wish  to  make  a  flattened  ovolo,  with  B 
as  a  centre  and  BA  as  radius,  describe  an  arc :  then,  with 
^  as  a  centre  and  ^J5  as  a  radius,  describe  a  second  arc, 
meeting  the  first  in  C.  Then,  with  C  as  a  centre,  describe 
the  arc  BA. 

12.  What  is  the  Bead  or  Astragal)  and  how  is  it  traced? 
■  It  is  a  thin  moulding,  of  which  the  circular  projection  is 

equal  to  half  the  height.  To  trace  it,  describe  a  semi-cir- 
cumference, of  which  the  diameter  AB  represents  the  height 
of  the  moulding. 

13.  What  is  a   Torus ^  and  how  traced? 

It  is  a  moulding  similar  to  the  bead,  but  thicker.     It  is 


OF    ARCHITECTURE.  ^9 

traced  by  describing  a    semi-circumference   on  the  height 
AB  as  a  diameter. 

14.  What  ts  a   Cavetto,  and  how  is  it  traced? 

A  cavetto  is  an  ovolo,  of  which  the  centre  C  is  in  a  per- 
pendicular from  the  extreme  projection  of  the  moulding.  It 
is  traced  by  describing  the  quarter  of  a  circumference  from  C 
as  a  centre.     The  second  figure  presents  a  cavetto  reversed. 

15.  What  is  a  Scotia^  and  how  is  it  traced  ? 

It  is  a  hollow  moulding,  formed  by  several  cavettos  with 
different  centres.  The  second  figure  represents  a  reversed 
scotia.  The  circular  parts  are  described  with  the  centres 
A  and  B. 

16.  What  is  a  Cj/ma-recta,  and  how  is  it  traced? 

The  cyma-recta  is  composed  of  an  o\^lo  and  a  cavetto. 
To  describe  it,  draw  the  line  AB,  and  then  divide  the  pro- 
jection of  the  moulding  into  two  equal  parts  by  the  perpen- 
dicular CD,  and  prod^uce  the  line  B :  the  poiilt  D  will,  be 
the  centre  of  the  ovolo,  and  the  point  C  of  the  cavetto, 
which  together  form  the  cyma-recta. 

The  flattened  cyriia-recta  is  a  similar  moulding.  To 
trace  it,  it  is  necessary,  after  having  divided  the  line  ^5 
into  two  equal  parts,  to  construct  an  equilateral  triangle  on 
each  of  the  parts.  The  points  C  and  D  will  then  be  the 
centres  of  the  arcs  which  form  the  moulding. 

17.  What  is  the  Ogee^  and  how  is  it  traced? 

The  ogee  is  a  moulding  composed  of  the  same  parts  as  the 
talon,  but  differently  placed.  Having  joined  the  points  A  and 
B,  we  draw  through  the  middle  point  of  this  line  the  line 
CD,  parallel  to  the  fillets  A  and  B,  and  the  points  C  and  ,7), 
in  which  it  meets  the  perpendiculars,  are  the  centres  of  the 
arcs  which  form  the  moulding.  If  the  ogee  is  flattened,  the 
centres  are  the  vertices  A  and  B  of  the  equilateral  triangles, 
each  constructed  on  the  half  of  DC. 


100  BOOK    IV. SECTION    I. 

18.  How  do  you  trace  this  moulding,  when  its  projection 
exceeds  its  height  ? 

Having  joined  the  points  A-  and  5,  divide  it  into  two 
equal  parts  AC,  BC,  and  then  draw  IP  perpendicular  to 
CA  at  the  middle  point.  Next,  draw  LN  perpendicular  to 
B  at  the  middle  point,  but  in  a  contrary  direction.  Then 
draw  BN  perpendicular  to  the  fillet ;  after  which  draw  NC, 
and  produce  it  to  P :  then  P  and  N  will  be  the  centres  of 
the  arcs.  To  give  grace  to  this  moulding,  the  part  BC  is 
sometimes  made  shorter  than  the  part  CA :  in  every  other 
respect  the  construction  is  the  same. 

19.  How  are  these  mouldings  to  be  used  in  combination  ? 

They  are  not  to  be  used  at  liazard,  ^ch  having  a  par- 
ticular situation  to  which  it  is  adapted,  and  where  it  must 
always  be  placed.  Thus,  the  ovolo  and  talon,  from  their 
peculiar  form,  seem  designed  to  support  other  important 
mouldings  ;  the  cyma  and  cavetto,  being  of  weaker  form, 
should  only  be  used  for  the  cover  or  shelter  of  the  other 
parts.  The  torus  and  astragal,  bearing  a  resemblance  to 
a  rope,  appear  calculated  to  bind  and  fortify  the  parts  to 
which  they  are  applied ;  while  the  use  of  the  fillet  and  scotia 
is  to  separate  one  moulding  from  another,  and  to  give  a 
variety  to  the  general  appearance. 

The  ovolo  and  cyma  are  mostly  placed  in  situations  above 
the  level  of  the  eye  :  when  placed  below  it,  they  should  only 
be  applied  to  crowning  members.  The  place  of  the  scolia 
is  universally  below  the  level  of  the  eye.  When  the  fillet 
is  very  wide,  and  used  under  the  cyma  of  a  cornice,  it  is 
called  a  corona ;  if  under  a  corona,  it  is  called  a  band. 

The  curved  contours  of  mouldings  are  portions  either  of 
circles  or  ellipses :  the  Greeks  always  preferred  the  latter. 


Fillet 


Plate-band 


,<■ 

Ovolo 

r 

\]S 

B 

Bead          j) 

A     . 

Scolia         I   B 


Scotia  (""g 


Oaee  or  Talon 


..P 


IB 


Ogee         \    \     \y^'''\^ 


Vn 

\   '- 

Ogee 

4  ^ 

N 


N 


102  BOOK    IV. SECTION    II. 


SECTION  II. 

OF    THE    ORDERS    OF     ARCHITECTURE,    AND    THEIR    PRINCIPAL 
PARTS. 

.^v^^.       1.  How  many  orders  of  architecture  are  there  ? 

Five :  the  Tuscan,  the  Doric,  the  Ionic,  the  Corinthian, 
and  the  Composite. 

2.  How  many  parts  do  we  distinguish  in  each  of  the  five 
orders  ? 

Three  :  the  pedestal,  the  column,  and  the  entablature. 

3.  Of  how  many  parts  is  the  pedestal  composed? 
Three :  the  plinth,  the  die,  and  the  cornice. 

4.  Of  how  many  parts  is  the  column  composed? 
Three:  the  base,  the  shaft,  and  the  capital. 

5.  Of  how  many  parts  is  the  entablature  composed  ? 
Three:  the  architrave,  the  frieze,  and  the  cornice. 

6.  Are  these  three  principal  parts  always  found  in  each  of 
the  orders? 

Not  always ;  for,  in  giving  the  name  of  an  order  to  an 
edifice,  regard  is  not  always  had  to  the  columns,  but  some- 
tames  to  the  proportions  observed  in  its  construction.  Some- 
times, even,  there  are  no  columns ;  and  often  the  pedestal 
is  replaced  by  the  plinth  only. 

7.  How  are  the  five  orders  distinguished  ? 

The  Tuscan  is  distinguished  by  the  simplicity  of  its  mem- 
bers, having  no  ornamient ;  the  Doric  by  the  triglyphs  which 
ornament  its  frieze  ;  the  Ionic  by  the  volutes  of  its  capital ; 
the  Corinthian  by  the  leaves  which  ornament  its  capital; 


Balustrade 


104  BOOK    IV. SECTION    II. 

and  the  Composite  by  the  Corinthian  capital,  united  with  the 
vohites  of  the  Ionic. 

8.  What  proportion  exists  between  the  diameter  and  height 
of  the  columns  in  the  different  orders? 

In  the  Tuscan  order,  the  height  of  the  column,  including 
its  base  and  capital,  is  seven  times  the  diameter  of  the  shaft 
at  the  base  ;  that  of  the  Doric  column  eight  times  ;  that  of 
the  Ionic  nine  times ;  and  that  of  the  Corinthian  and  Com» 
posite  ten  times. 

9.  What  proportions  are  established  between  the  three  jmn- 
cipal  parts,  in  the  orders  of  architecture  ? 

In  all  the  orders,  the  pedestal  is  one  third  the  height  of 
the  column,  and  the  entablature  is  one  quarter  the  height, 

10.  What  is  a  module  ?  > 

In  all  the  orders  except  the  Doric,  it  is  the  diameter  of 
the  shaft  at  the  base :  in  the  Doric,  it  is  the  semi-diameter. 
This  is  according  to  Gwilt's  Architecture,  published  at 
London  in  1839.  Some  authors  call  the  semi-diameter  a 
module. 

11.  What  is  a  minute? 

In  the  Doric  order,  the  module  is  supposed  to  be  divided 
into  thirty  equal  parts,  and  in  each  of  the  other  orders  into 
sixty,  and  each  of  the  equal  parts  is  called  a  minute  :  hence 
a  minute  is  one-sixtieth  part  of  the  diameter  of  the  shaft 
at  its  base. 

12.  If  the  diameter  of  a  shaft  is  two  feet  at  the  base,  what 
will  be  the  height  of  the  structure  in  each  of  the  five  orders? 

FOR    THE    TUSCAN    ORDER. 

2  X  7  =  14  feet    —  height  of  column,  (Art.  8) 
add  one-third    =    4ft.  Sin.  =  height  of  pedestal,  (Art.  9) 
add  one-fourth  =    3ft.  6in.  =  height  of  entablature,  (Art.  9.) 

Total  height      ^  22ft.  2in. 


106  BOOK    IV. SECTION    II. 

By  a  similar  process,  we  should  find  the  height  of  the 
Doric  to  he  25  feet  4  inches  ;  that  of  the  Ionic,  28  feet 
6  inches  ;  that  of  the  Corinthian,  31  feet  8  inches  ;  and  that 
of  the  Composite,  31  feet  8  inches. 

13.  What  is  the  form  of  the  shafts  of  the  columns? 
The  shafts  diminish  in  diameter  as  they  rise :  sometimes 

the  tapering  begins  at  the  foot  of  the  shaft ;  sometimes  from 
a  point  one  quarter  from  the  base,  and  sometimes  from  a 
point  one-third ;  and  in  some  examples  there  is  a  swelling 
in  the  middle.  The  difference  between  the  diameter  at  top 
and  that  at  bottom,  is  seldom  more  than  one-sixth  of  the 
least  diameter,  or  less  than  one -eighth. 

1 4.  What  do  you  remark  of  the  entablatures  7 

The  entablature  and  its  subdivisions,  though  architects 
frequently  vary  from  the  proportions,  may  as  a  general  rule 
be  set  down  as  exhibited  in  the  drawings.  The  total  height 
of  the  entablature,  in  all  the  orders  except  the  Doric,  is 
divided  into  10  parts,  three  of  which  are  given  to  the  archi- 
trave, three  to  the  frieze,  and  four  to  the  cornice.  But  in 
the  Doric  order,  the  whole  should  be  divided  into  eight, 
parts,  two  given  to  the  architrave,  three  to  the. frieze,  and 
three  to  the  cornice.  The  mouldings,  which  form  the  de- 
tail of  these  leading  features,  are  best  learned  by  reference 
to  representations  of  the  orders  at  large. 

In  the  Ionic  order  the  entablatures  are  generally  very 
simple  The  architrave  has  one  or  two  fasciae;  the  frieze 
is  plain,  and  the  cornice  has  four  parts.  In  the  Composite 
order,  the  entablature  is  large  for  so  slender  an  order ;  yet 
it  is  on  many  accounts  very  beautiful. 


MENSURATION    OF    SURFACES. 


109 


BOOK   V. 


SECTION   I. 


MENSURATION    OF    SURFACES. 


1.   What  do  you  understand  by  the  unit  of  length? 

If  the  length  of  a  line  be  computed  in  feet,  one  foot  is 
the  unit  of  the  line,  and  is  called  the  linear  unit. 

if  the  length  of  a  line  be  computed  in  yards,  one  yard 
is  the  linear  unit.  If  it  be  computed  in  rods,  one  rod  is 
the  linear  unit;  and  if  it  be  computed  in  chains,  one  chain 
is  the  linear  unit. 


2.  What  do  you  understand  hy  the  unit  of  surface  ? 


If  we  describe  a  square  on  the  unit 
of  length,  such  square  is  called  the  unit 
of  surface.  Thus,  if  the  linear  unit  be 
1  foot,  one  square  foot  will  be  the  unit 
of  surface.  / 


3.  How  many  square  feet  are  there  in 
a  square  yard? 

If  the  linear  unit  is  1  yard,  one  square 
yard  will  be  the  unit  of  surface  ;  and  this 
square  yard  colitains  9  square  feet.    . 


1  yard  =  3  feet. 

no 


BOOK    V. SECTION    I. 


1  chain  =  4  rods. 


4.  How  many  square  rods  are  there 
in  a  square  chain  ? 

If  the  linear  unit  is  1  chain,  the 
unit  of  surface  will  be  1  square  chain, 
whicTi  will  contain  1 6  square  rods. 


5.  How   do  you  find  the  number  of  square  feet  contained 
in  a  rectangle  ? 

If  we  have  a  rectangle  whose  base 
is  4  feet,  and  altitude  3  feet,  it  is  evi- 
dent that  it  will  contain  12  square 
feet.  These  12  square  feet  are  the 
measure  of  the  surface  of  the  rectan- 
gle. 


6.  How  do  you  find  the  number  of  squares  contained  in 
any  rectangle  ? 

.  It  is  ^lain  that  ttie  number  of  squares  in  any  rectangle, 
\yill  be  expressed  by  the  units  of  its  base,  multiplied  by  the 
units  in  its  altitude.  This.,  product  is  called  the  measure 
of  the  rectangle 

7.  What  do  you  mean  by  the  rectangle  of  two  lines? 

In  geometry,  we  often  say,  the  rectangle  of  two  lines, 
by  which  we  mean,  the  rectangle  of  which  those  lines  are 
the  two  adjacent  sides. 

8.  What  is  the  area  of  a  figure  ? 
The  measure  of  its  surface. 


9.   What   is   the  unit   of  the   number  which    expresses   the 
area  ? 

It  is  a  square,  of  which  the  linear  unit  is  the  side. 


MENSURATION  OF  SURFACES. 


Ill 


to.  How  do  ypu  find  the  area  of  a  rectangle? 

The  area  of  a  rectangle  is  equal  to  the  product  of  its 
base  by  its  altitude.  If  the  base  of  a  rectangle  is  30  yards, 
and' the  altitude  5  yards,  the  area  will  be  150  square  yards. 

11.  What  is  the  area  of  a  square  equal  to? 

The  area  of  a  square  is  equal  to  the  product  of  its  two 
equal  sides ;  that  is,  to  the  square  of  one  of  its  sides. 

12.  How  does  the  diagonal  of  a  rectangle  divide  it? 
The  diagonal  DB  divides  the  rect-        D C 

angle  ABCD  into  two  equal  triangles. 
Hence,  a  triangle  is  half  a  rectangle, 
having  the  same  base  and  altitude. 

13.  What  is  the  altitude  of  a  parallelogram? 
The  altitude  of  a  parallelogram  is 

the  perpendicular  distance  between 
two  of  its  parallel  sides.  Thus,  EB 
is  the  altitude  of  the  parallelogram 
ABCD. 

14.  What  part  of  a  parallelogram  is  a  triangle,  having  the 
same  base  and  the  same  altitude  ? 

A  triangle  is  also  half  a  parallelo- 
gram, having  the  same  base  and  alti- 
tude. 

15.  What  is  the  area  of  a  triangle  equal  to  ? 
The  area  of  a  triangle  is  equal  to 

half  the  product  of  the  base  by  the 
altitude ;  for,  the  base  multiplied  by 
the  altitude  gives  a  rectangle  which 
is  double  the  triangle.  Thus,  the  area 
of  the  triangle  ABC  is  equal  to  half 
the  product  of  AB  x  CD. 


112  BOOK    V. SECTION    1. 

If  the  base  of  a  triangle  is  12,  and  the  altitude  8  yards, 
the  area  will  be  48  square  yards. 

16.  What  is  the  area  of  a  parallelogram  ? 
The   area    of  a    parallelogram   is 

equal  to  its  base  multiplied  by  its 
altitude.  Thus,  the  area  of  the  par- 
allelogram ABCD  is  equal  to  AB 
X  BE.  -^ 

If  the  base  is  20,  and  altitude  15  feet,  the  area  will  be 
300  square  feet. 

17.  What  is  the  area  of  a  trapezoid? 

The   area  of  a  trapezoid  is  equal  to  jr)  q 

half  the  sum  of  its  parallel  sides  multi-         / 

plied  by  the  perpendicular  distance  be-  Z 

tween  them.     Thus, 

area  ABCD  =  x{AB  -f  CD)  X  CF. 


F  B 


18.  With  what  is  land  generally  measured  ? 
Surveyors,  in   measuring   land,    generally  use   a  chain, 

called  Gunter's  chain.     This  chain  is  four  rods,  or  66  feet 
in  length,  and  is  divided  into  100  links. 

19.  What  is  an  acre? 

An  acre  is  a  surface  equal  in  extent  to  10  square  chains ; 
that  is,  equal  to  a  rectangle  of  which  one  side  is  ten  chains, 
and  the  other  side  one  chain. 

20.  What  is  a  quarter  of  an  acre  called  ? 
One  quarter  of  an  atire  is  called  a  rood. 

21.  How  many  square  rods  in  an  acre? 

Since  the  chain  is  4  rods  in  length,  1  square  chain  con- 
tains 16  square  rods;  and  therefore,  an  acre,  which  is  10 
square  chains,  contains  160  square  rods,  and  a  rood  con- 
tains 40  square  rods.     The  square  rods  are  called  perches. 


MENSURATION  OF  SURFACES.  113 

22.  How  is  land  generally^  computed  ? 

Land  is  generally  computed  in  acres,  roods,  and  perches, 
which  are  respectively  designated  by  the  letters  A.  R.  P. 

23.  If  the  linear  dimensions  are  chains  or  links,  how  do 
you  find  the  acres  ? 

When  the  linear  dimensions  of  a  survey  are  chains  or 
links,  the  area  will  be  expressed  in  square  chains  or  square 
links,  and  it  is  necessary  to  form  a  rule  for  reducing  this 
area  to  acres,  roods,  and  perches.  For  this  purpose,  let 
us  form  the  following 

TABLE.       '    , 

1  square  chain  =  10000  square  links. 
1  acre  =  10  square  chains  =  100000  square  links. 

1  acre  =  4  roods  =  160  perches. 
1  square  mile  =  6400  square  chains  =  640  acres. 

24.  If  the  linear  dimensions  are  links,  how  do  you  find  the 
acres  ? 

When  the  linear  dimensions  are  links,  the  area  will  be 
expressed  in  square  links,  and  may  be  reduced  to  acres  by 
dividing  by  100000,  the  number  of  square  links  in  an  acre  : 
that  is,  by  pointing  off  five  decimal  places  from  the  right 
hand. 

If  the  decimal  part  be  then  multiplied  by  4,  and  five 
places  of  decimals  pointed  oflf  from  the  right  hand,  the  fig- 
ures to  the  left  will  express  the  roods. 

If  the  decimal  part  of  this  result  be  now  multiplied  by 
40,  and  five  places-  for  decimals  pointed  off,  as  before,  the 
figures  to  the  left  will  express  the  perches. 

If  one  of  the  dimensions  be  in  links,  and  the  other  in 
chains,  the  chains  may  be  reduced  to  links  by  annexing 
two  ciphers :  or,  the  multiplication  may  be  made  without 
annexing  the  ciphers,  and  the  product  reduced  to  acres  and 


114  BOOK    V. SECTIO*?!    I. 

decimals  of  an  acre,  by  pointing  off  three  decimal  places 
at  the  right  hand. 

Whenboth  the  dimensions  are  in  chains,  the  product  is 
reduced  to  acres  by  dividing  by  10,  or  pointing  off  one  deci- 
mal place. 

From  which  we  conclude  that, 

1st.  If  links  be  multiplied  by  links,  the  product  is  reduced 
to  acres  by  pointing  off  five  decimal  places  from  the  right 
hand. 

2d.  If  chains  be  multiplied  by  links,  the  product  is  re- 
duced to  acres  by  pointing  off  three  decimal  places  from 
the  right  hand, 

3d.  If  chains  be  multiplied  by  chains,  the  ^  product  is  re- 
duced to  acre^by  pointing  off  one  decimal  place  from  the 
right  hand. 

25.  How  do  you  find  the  number  of  square  feet  in  an 
acre? 

Since  there  are  16.5  feet  in  a  rod,  a  square  rod  is 
equal  to 

16.5  X  16.5  =  272.25  square  feet. 
If  the  last  number  be  multiplied  by  160;  the  number  of 
square  rods  in  an  acre,  we  shall  have 

272.25  X  160  =  43560  =  the  squ^are  feet  in  an  acre. 

OF    THE    TRIANGLE. 

26.  How  do  you  find  the  area  of  a  triangle,  when  the  base 
and  altitude  are  known  ? 

1st.  Multiply  the  base  by  the  altitude,  and  half  the  product 
will  be  the  area. 

Ot,  2d.  Multiply  the  base  by  half  the  altitude,  and  the 
product  will  be  the  area. 


MENSURATION    OF    SURFACES. 


115 


EXAMPLES. 


I.  Required  the  area  of  the  triangle 
ABC^  whose  base  AB  is  10.75  feet  and 
altitude  7.25  feet. 


We  first  multiply  the  base 
by  the  altitude,  and  then  di- 
vide the  product  by  2. 


Operation. 
10.75  X  7.25  =:  77.9375 
and 
77.9375  -^  2  =  38.96875 


2.  What  is  the  area  of  a  triangle  whose  base  is  18  feet 
4  inches,  and  altitude   11   feet  10  inches? 

Ans.   108  sq.ft   5'  8'\ 

3.  What  is  the  area  of  a  triangle  whose  base  is  12.25 
chains,  and  altitude  8.5  chains? 

Ans.  5  A.  OR.  33  P. 

4.  What   is  the   area   of  a   triangle   whose  base    is  20 
feet,  and  altitude  10.25  feet? 

Ans.  102.5  sq.  ft. 

5.  Find  the  area  of  a  triangle  whose  base  is  625  and 

altitude  520  feet 

Ans.   162500  sq.ft. 

6.  Find  the  number  of  square  yards  in  a  triangle  whose 
base  is  40  and  altitude  30  feet. 

A71S.  66|  sq.  yds, 

7.  What  is  the   area  of  a  triangle  whose  base  is  72.7 
yards,  and  altitude  36.5  yards  ? 

Ans.  1326.775  sq.  yds. 

8.  What  is  the  content  of  a  triangular  field  whose  base 
is  25.01   chains,  and  perpendicular  18.14  chains? 

Ans.  22  A.  2  R.  29  P. 


116  BOOK    V. SECTION    I. 

9.  What  is  the  content  of  a  triangular  field  whose  base 
is  15.48  chains,  and  altitude  9.67  chains? 

Ans.  7  A.  I  R.  38  P. 

27.  How  do  you  find  the  area  bf  a  triangle  when  the  three 
sides  are  given  ? 

1st.  Add  the  three  ^ides  together  and  take  half  their  sum. 

2d.  From  this  half  sum  take  each  side  Separately. 

3d.  Multiply  together  the  half  sum  and  each  of  the  three 
remainders,  and  then  extract  the  square  root  of  the  product, 
which  will  be  the  required  area. 

EJCAMPLES.  '  ^ 

1.  Find  the  area  of  a  triangle  whose  sides  are  20,  30, 

and  40  rods. 

20  45  45  45 

30  20 30       40 

^^  25  1st  rem.      15  2d  rem.         5  3d  rem. 

2)90  • 

45  half  sum. 


Then,  to  obtain  the  product,  we  have 

45  X  25  X  15  X  5  =  84375  ; 
from  which  we  find 

area  ^  V84375  =  290.4737  perches. 

2.  How  many  square  yards  of  plastering  are  there  in  a 
triangle,  whose  sides  are  30,  40,  and  50  feet? 

Ans.  66|. 

3.  The  sides  of  a  triangular  field  are  49  chains,  50.25 
chains,  and  25.69  chains :  what  is  its  area  ? 

Ans.  61  A,  1  R.  39.68  P. 

4.  What  is  the  area  of  an  isosceles  triangle,  whose  base 
is  20,  and  each  of  the  equal  sides  15? 

Ans.  111.803. 


MENSURATION    OF    SURFACES. 


117 


5.  How  many  acres  are  there  in  a  triangle  whose  three 
sides  are  380,  420,  and  765  yards? 

Alls.  9  A.  0  R.  38  P. 

6.  How  many  square  yards  in  a  triangle  whose   sides 

are  13,  14,  and  15  feet? 

Ans.  91. 

7.  What  is  the  area  of  an  equilateral  triangle  whose  side 

is  25  feet? 

Ans.  270.6329  sq.ft. 

8.  What  is  the   area  of  a  triangle  whose  sides  are  24, 

36,  and  48  yards? 

Ans.  418.282  sq.  yds. 

28.  How  do  you  find  the  hypothenuse  of  a  right-angled 
triangle  when  the  base  and  perpendicular  are  known  ? 

1st.  Square  each  of  the  sides  separately. 

2d.  Add  the  squares  together. 

3d.  Extract  the  square  root  of  the  sum,  which  will  be 
the  hypothenuse  of  the  triangle. 


EXAMPLES. 

1.  In  the  right-angled  triangle  ABC,  we 
have 

AB  =  30  feet,  BC  =  40  feet, 
to  find  ^C. 


We  first  square  each  side, 
and  then  take  the  sum,  of 
which  we  extract  the  square 
root,  which  gives 

AC  =  1/2500  =z  50  feet. 


Operation. 
30'  =    900 
40*  =  1600 


sum  =  2500 


2.  The  wall  of  a  building,  on  the  brink  of  a  river,  is  120 
feet  high,  and  the  breadth  of  the  river  70  yards :  what  is 


118 


BOOK    V. SECTION    I. 


the  length  of  a  line  which  would  reach  ftom  the  top  of  the 
wall  to  the  opposite  edge  of  the  river  ? 

Ans.  241.86  ft. 

3.  The  side  roofs  of  a  house  of  which  the  eaves  are  of 
the  same  height,  form  a  right  angle  at  the  top.  Now,  the 
length  of  the  rafters  on  one  side  is  10  feet,  and  on  the  other 
14  feet :  what  is  the  breadth  of  the  house  ? 

Ans.  17.204  ft. 

4.  What  would  be  the  width  of  the  house,  in  the  last 
example,  if  the  rafters  on  each  side  were  10  feel? 

Ans.  14.142/^. 

5.  What  would  be  the  width,  if  the  rafters  on  each  side 
were  14  feet  ? 

Ans.   19.7989/^. 

29.  If  the  hypothenuse  and  one  side  of  a  right-angled  tri- 
angle are  known,  how  do  you  find  the  other  side? 

Square  the  hypothenuse  and  also  the  other  given  side, 
and  take  their  difference :  extract  the  square  root  of  this 
difference,  and- the  result  will  be  the  required  side. 


EXAMPLES. 

1.  In   the   right-angled   triangle  ABC, 
there  are  given 

AC  =  50  feet,  and  AB  =  40  feet; 
required  the  side  BC. 

We  first  square  the  hypo- 
thenuse and  the  other  side, 
after  which  we  take  the  dif- 
ference; and  then  extract  the 
square'  root,  which  gives 

BC  =  V90b  =  30  feet 


Operation. 

50' 

=  2500 

40' 

=  1600 

Diff 

=    900 

AREA    OF    THE    SQUAllE.  119 

2.  The  height  of  a  precipice  on  the  brink  of  a  river  is 
103  feet,  and  a  line  of  320  feet  in  length  will  just  reach 
from  the  top  of  it  to  the  opposite  bank :  required  the  breadth 
of  the  river. 

A71S.  302.9703//. 

3.  The  hypothenuse  of  a  triangle  is  53  yards,  and  the 
perpendicular  45  yards  :  what  is  the  base  ? 

Ans.  28  yds. 

4.  A  ladder  60  feet  in  length,  will  reach  to  a  window 
40  feet  from  the'  ground  on  one  side  of  the  street,  and  by 
turning  it  oVer  to  the  other  side,  it  will  reach  a  window  50 
feet  from  the  ground:   required  the  breadth  of  the  street. 

Ans.  77.8875/. 

AREA  OF  THE  SQUARE. 

30.  How  do  you  find  the  area  of  a  square,  a  rectangle^  or 
a  parallelogram  ? 

Multiply  the  base  by  the  perpendicular  height,  and  the 
product  will  be  the  area. 

r. 

1.     Required   the    area    of    the    square 
ABCD,  each  of  whose  sides  is  36  feet. 


Operation. 
36  X  36=  1296  sq.ft. 


We  multiply  two  sides  of 
the  square  together,  and  .  the 
product  is  the  area  in  square 
feet. 

2.  How  many   acres,   roods,   and  perches,  in   a   square 
whose  side  is  35.25  chains? 

Ans.  124  A.  I  R.  I  P. 

3.  What  is  the  area  of  a  square  whose  side  is  8  feet 
4  inches?     (See  Arithmetic,  §  171.) 

,  ,     -  Ans.  69  ft.  5'  4f'^ 


120 


BOOK    V. SECTION    I. 


4.  What  is  the  content  of  a   square  field  whose  side  is 

46  rods  ?  * 

Ans.  13  A.  0  R.  36  P. 

5.  What  is  the    area  of  a  square  whose   side  is   4769 
yards  ? 

Ans.  22743361  sq.  yds. 

AREA    OF    THE    RECTANGLE. 


1.  To  find  the  area  of  a  rectangle 
ABCDj  of  which  the  base  AB  =  45 
yards,  and  the  altitude  AD  =  15  yards. 


Here  we  simply  multiply 
the  base  by  the  altitude,  and 
the  product  is  the  area. 


Operation. 
45  X  15  =  675  sq.  yds. 


2.  What  is  the   area   of  a  rectangle  whose  base  is  14 
feet  6  inches,  and  breadth  4  feet  9  inches? 

Ans.  68  sq.ft.  10' 6^^ 

3.  Find  the  area  of  a  rectangular  board  whose  length  is 
112  feet,  and  breadth  9  inches. 

Ans.  84  sq.ft. 

4.  Required  the  area  of  a  rectangle  whose  base  is  10.51, 
lind  breadth  4.28  chains. 

Ans.  4  A.  I  R.  39.7  P +. 

5.  Required  the  area  of  a  rectangle  whose  base  is  12 
feet  6  inches,  and  altitude  9  feet  3  inches. 

Ans.   115  sq.ft.  T  6'^ 


AREA    OF    THE    PARALLELOGRAM. 


1.  What  is  the  area  of  the  paral- 
lelogram ABCD,  of  which  the  base 
AB  is  64  feet,  and  altitude  DE,  36 
feet? 


AREA    OF    THE    TRAPEZOID. 


121 


Operation. 
64  X  36  =  2304  sq.  ft. 


We  multiply  the  base  64,  1 
by  the  perpendicular  height 
36,   and  the   product  is   the 
required  area. 

2.  What  is  the  area  of  a  parallelogram  whose  base  is 
12.25  yards,  and  altitude  8.5  yards  ? 

Alls.   104.125  sq.  yds. 

3.  What  is  the  area  of  a  parallelogram  whose  base  is 
8.75  chains,  and  altitude, 6  chains? 

Ans.  5  A.  1  R.  0  P. 

4.  What  is  the  area  of  a  parallelogram  whose  base  is 
7  feet  9  inches,  and  altitude  3  feet  6  inches  ? 

Ans.  27  sq.ft.  V  6''. 

5.  What  is  the  area  of  a  parallelogram  whose  base  is 
10.50  chains,  and  breadth  14.28  chains? 

Ans.  14  A.  3  R.  30P-f. 

AREA    OF    THE    TRAPEZOID- 

31.  How  do  you  find  the  area  of  a  trapezoid? 

Multiply  the  sum  of  the  parallel  sides  by  the  perpendicu- 
lar distance  between  them,  and  then  divide  the  product  by 
two :  the  quotient  will  be  the  area. 


EXAMPLES. 


1.  Required  the  area  of  the  trapezoid 
ABCD,  having  given 


D 


E 


AB  rr:  321.51  ft.,  DC 


We  first  find  the  sum  of 
the  sides,  and  then  multi- 
ply it  by  the  perpendicular 
height,  after  which,  we  di- 
vide the  product  by  2,  for 
the  area. 


214.24  ft,,  and  CE  =  171.16  ft. 
Operation. 
321.51+214,24=535.75- 
sum  of  parallel  sides. 

Then, 
535.75  X  171.16  =  91698.97 


and, 


91698.97 


=  45849.485- 


=  the  area. 


122  BOOK    V. SECTION    I. 

2.  What  is  the  are?i  of  a  trapezoid,  the  parallel  sides  of 
which  are  12.41  and  8.22  chains,  and  the  perpendicular 
distance  between  them  5.15  chains? 

Ans.  5  A.  1  R.  9.956  P. 

3.  Required  the  area  of  a  trapezoid  whose  parallel  sides 
are  ^5  feet  6  inches,  and  1  8  feet  9  inches,  and  the  per- 
pendicular distance  between  them  10  feet  and  5  inches. 

Ans.  230  sq.ft.  5'  T' . 

4.  Required  the  area  of  a  trapezoid  whose  parallel  sides 
are  20.5  and  12.25,  and  the  perpendicular  distance  between 
them  10.75  yards. 

Ans.   176.03125  sq.  yds. 

5.  What  is  the  area  of  a  trapezoid  whose  parallel  sides 
are  7.50  chains,  and  12.25  chains,  and  the  perpendicular 
height  15.40  chains? 

Ans.  15  A.  0  R.  32.2  P. 

6.  What  is  the  content  when  the  parallel  sides  are  20 
and  32  chains,  and  the  perpendicular  distance  between  them 
26  chains? 

Ans.  Ql  A.  2  R.  \QP. 

A^EA    OF    A    QUADRILATERAL. 

32.  How  do  you  find  the  area  of  a  quadrilateral  ? 

Measure  the  four  sides  of  a  quadrilateral,  and  also  one 
of  the  diagonals :  the  quadrilateral  will  thus  be  divided  into 
two  triangles,  in  both  of  which  all  the  sides  will  be  known. 
Then,  find  the  areas  of  the  triangles  separately,  and  their 
sum  will  be  the  area  of  the  quadrilateral. 

EXAMPLES. 

D 
1 .  Suppose  that  we  hate  meas-  y^  i  \ 

ured  the  sides  and  diagonal  AC,  i  ,    ^\ 

of  the  quadrilateral  ABCD,  and     ^"^^        \ 
found  ^\ 


AREA    OF    A    QUADRILATERAL.  123 

AB  =  40.05  ch,    CD  =  29.87  ch, 
BC  =  26.27  ch,    AD  =  37,07  ch, 

and  AC  —  55  ch: 

required  the  area  of  the  quadrilateral. 

Ans.  101  A.  1  R.  15  P. 

Remark. — Instead  of  measuring  the  four  sides  of  the 
quadrilateral,  we  may  let.,  fall  the  perpendiculars  Bh,  Dg, 
on  the  diagonal  jIC.  The  area  of  the  triangles  may  then 
be  determined  by  measuring  these  perpendiculars  and  the 
diagonal  AC.  The  perpendiculars  are  Dg  —  18.95  ch,  and 
Bb  =  17.92  ch. 

2.  Required  the  area  of  a  quadrilateral  whose  diagonal 
\s  80.5  and  two  perpendiculars  24.5  and  30.1   feet. 

Ans.  2197.65  sq.ft. 

3.  What  is  the  area  of  a  quadrilateral  whose  diagonal  is 

108  feet  6  inches,  and  the  perpendiculars  56  feet  3  inches, 

and  60  feet  9  inches  ? 

Ans.  6347  sq.ft.  3^ 

4.  How  many  square  yards  of  paving  in  a  quadrilateral 
whose  diagonal  is  65  feet,  and  the  two  perpendiculars  28 
and  331  feet? 

Ans.  222y'j  sq.  yds. 

5.  Required  the  area  of  a  quadrilateral,  whose  diagonal 
is  42  feet,  and  the  two  perpendiculars  18  and  16  feet. 

Ans.  714  sq.ft. 

6.  What   is    the    area  of  a  quadrilateral    in   which   the 

diagonal  is  320.75  chains,  and  the  two  perpendiculars  69.73 

chains,  and  130.27  chains? 

Ans.  3207  A.  2  R. 

33.  How  do  you  find  the  area  of  a  long  and  irregular  Jig- 
ure,  bounded  on  one  side  by  a  straight  line  ? 

1st.   Divide   the  right  line   or  base   into  any  number  of 


124 


BOOK    V. SECTION    I. 


equal  parts,  and  measure  the  breadth  of  the  figure  at  the 
points  of  division,  and  also  at  the  extremities  of  the  base. 

2d.  Add  together  the  intermediate  breadths,  and  half  the 
sum  of  the  extreme  ones. 

3d.  Multiply  this  sum  by  the  base  line,  and  divide  the 
product  by  the  number  of  equal  parts  of  the  base. 

EXAMPLES. 

1.  .The  brea^dths  of  an  irregular 
figure,  at  five  equidistant  places, 
A,  B,  C,  D,  and  .E,  being  8.20  ch, 
7.40  ch,  9.20  ch,  10.20  ch,  and 
8.60  chains,  and  the  whole  length  40  chains ;  required  the 
area. 

8.20  35.20 

8.60  40 

2)16.80 

8.40  mean  of  the  extremes. 
7.40 
9.20 
10.20 


4)1408.00 

352.00  square  ch. 


35.20  sum. 


Ans.  35  A.  32  P. 

2.  The  length  of  an  irregular  piece  of  land  being  21  ch, 
and  the  breadths,  at  six  equidistant  points,  being  4.35  ch, 
5.15  ch,  3.55  ch,  4.12  ch,  5.02  ch,  and  6.10  chains:  re- 
quired the  area.  Ans.  9  A.  2  R.  30  P. 

3.  The  length  of  an  irregular  figure  is  84  yards,  gtod  the 
breadths  at  six  equidistant  places  are  17.4 ;  20.6 ;  14.2  ; 
16.5  ;  20.1,  and  24.4:  what  is  the  area? 

Ans.   1550.64  sq.  yds. 

4.  The  length  of  an  irregular  field  is  39  rods,  and  its 
breadths  at  five  equidistant  places  are  4.8 ;  5.2  ;  4.1  ;  7.3, 
and  7.2  rods :  what  is  its  area  ? 

Ans.  220.35  sq:  rods. 


OF    POLYGONS, 


125 


5.  The  length  of  an  irregular  field  is  50  yards,  and  its 
breadths  at  seven  equidistant  points  are  5.5  ;  6.2;  7.3;  6; 
7.5 ;  7 ;  and  8.8  yards  :  what  is  its  area  ? 

Ans.  342.916  sq.  yds. 

6.  The  length  of  an  irregular  figure  being  37.6,  and  the 
breadths  at  nine  equidistant  places,  0  ;  4.4  ;  6.5  ;  7.6  ;  5.4  ; 
8';  5.2;  6.5;  and  6.1  :  what  is  the  area? 

Ans.  219.255. 


OF    POLYGONS. 

34.    What  is  a  regular  Polygon  ? 
A  regular  polygon  is    one   which 
has  all  its  sides  equal  to  each  other, 
eacli  to  each,  alid  all  its  angles  equal 
to  each  other,  each  to  each. 

Thus,  if  the  polygon  ABCDE  be 
regular,  we  have 

AB  =  BC  =  CD  =  DE  =  EA : 
angle  A  =  B  =  C  =  D  =  E. 


also 


35.    What  are  similar  polygons  ? 

Similar  polygons  are  those  which  have  the  angles  of 
the  one  equal  to  the  angles  of  the  other,  each  to  each, 
and  the  sides  about  the 
equal  angles  propor- 
tional. Hence,  similar 
polygons  are  alike  in 
shape,  but  may  differ 
in  size. 

The  sides  which  are 
like    situated    in    two 
similar  polygons,   are    called  homologous   sides,  and   these 
sides  are  proportional  to  each  other.  ^  ' 


126 


BOOK    V. — SECTION    I. 


Thus,     if    ABCDE 
and  FGHIK  are   two 
similar  polygons  :  then      „, 
angle   A  =.  F,  B  =  G,        ^ 
C  =  H,    D  =  I,     and 
E  =  K. 


Also,  AB  :  FG 

and  AB  :  FG 

also,  AB  :  FG 

and  AB  :  FG 


EC  :   GH 
CD  :  HI; 
DE  :  IK 
EA  :  KF. 


36.  Into  how  many  triangles  may  any  polygon  be  divided  1 

Any  polygon  may  be  divided  by  di- 
agonals, into  as  many  triangles  less 
two,  as  the  polygon  has  sides.  Thus, 
if  the  polygon  has  five  sides,  there  will 
be  three  triangles  ;  if  it  has  six  sides, 
there  will  be  four ;  if  seven  sides,  five ; 
if  eight  sides,  six  ;  &c. 


37.  What  is  the  sum  of  all  the  in- 
ward angles  of  a  polygon  equal  to  ? 

The  sum  of  all  the  inward  angles 
of  any  polygon  is  equal  to  twice  as 
many  right  angles,  wanting  four,  as 
the  figure  has  sides.  Thus,  if  the 
polygon  has  five  sides,  we  have 

A-\-  B  -^  C  -\-D  +  E=  10  right  angl 
=  6  right  angles. 


B 

4  right  angles 


38.  What  is  the  sum  of  the  angles  of  a  quadrilateral 
equal  to  ? 

If  the  polygon  is  a  quadrilateral,  then  the  sum  of  the 
angles  will  be  equal  to  four  right  angles. 


OP    POLYGONS.  127 

39.  How  do  y oil  find  one  df  the  angles  of  a  regular  poly- 
gon? 

When  the  polygon  is  regular,  its  angles  will  be  equal  to 
each  other.  If,  then,  the  sum  of  the  inward  angles  be  di- 
vided by  the  number  of  angles,  the  quotient  will  be  the  value 
of  one  of  the  angles.  We  shall  find  the  value  in  degrees, 
by  simply  placing  90°  for  the  right  angle. 

40.  How  do  you  find  one  of  the  angles  of  an  equilateral 
triangle  1 

The  «um  of  all  the  angles  of  an  equilateral  triangle  is 
equal  to 

6  X  90°  -  4  X  90O  =  540°  -  360°  =  ISO^ 
and  for  each  angle  , 

180° -f-  3  =  60°: 
Hence,  each  angle  of  an  equilateral  triangle  is  equal  to 
60  degrees. 

41.  How  do  you  find  one  of  the  angles  of  a  square  or 
rectangle  ? 

The  sum  of  all  the  angles  of  a  square  or  rectangle  is 
8  X  90°  —  4  X  90O  =:  720^^  —  360°  =  360°  : 
and  for  each  of  the  angles 

360°  -^  4  =  90°. 

42.  How  do  you  find  one  of  the  angles  of  a  regular  pen- 
tagon ? 

The  sum  of  all  the  angles  of  a  regular  pentagon  is 
equal  to 

10  X  90°  —  4  X  90O  =  900°  —  360°  =  540° : 
and  for  each  angle 

540°  -^-  5  =  108°. 

43.  How  do  you  find  one  of  the  angles  of  a  regular  hexa- 
gon ? 

The  sum  of  all  the  angles  of  a  regular  hexagon  is  equal  to 


128 


BOOK    V. SECTION    I. 

1080°  —  360O 


12  X  90°  —  4  X  90° 
and  for  each  angle 

720O  -^  6  =  120°. 


720°: 


44.  How  do  you  find  one  of  the  angles  of  a  Yegular  heptO' 
gon? 

The  sum  of  the  angles  of  a  regular  heptagon  is  equal  to 
14  X  90O  —  4  X  900  _  1260°  —  360^  ^  900°  : 
and  for  one  of  the  angles 

900°-^  7=  1280  34^+. 


45.  How  do  you  find  one  of  the  angles  of  a  regular  octa- 
gon ? 

The  sum  of  the  angles  of  a  regular  octagon  is  equal  to 
16  X  90°  —  4  X  90O  =  1440°  —  360°  =  1080° : 
and  for  each  angle 

1080°  -h  8  =  135°. 


46.  How  many  figures  can  he  arranged  about  a  point  so  as 
to  fill  up  the  entire  space? 

There  are  but  three ;  the  equilateral  triangle,  the  square 
or  rectangle,  and  the  hexagon. 


First. — Six  equilateral  triangles  placed 
about  the  point  C,  will  -fill  the  entire 
space.  For,  each  angle  is  equal  to  60°, 
and  their  sum  to 

60°  X  6  =  360°. 


Second. — Four  squares,  or  rectangles, 
placed  about  C,  will  fill  the  entire  space. 
For,  each  angle  is  equal  to  90°,  and 
th^ir  sum  to 

90°  X  4  =  360°. 


. 

c 

OF    POLYGONS. 


129 


Third.  —  Three  hexagons 
placed  about  C,  will  fill  up 
the  entire  space.  For,  each 
angle  is  equal  to  120°,  and 
the  sum  of  the  three  to 
nO^  X  3  =  360°. 


47.  How  are  similar  polygons  to  each  other? 
Similar  polygons  are 


7) 


vto  each  other  as  the 
squares  described  on 
their  homologous  sides. 
Thus,  the  two  simi- 
lar polygons  ABODE  J 
FGHIK,  are  to  each 
other  as  the  squares 
described  on  the  homo- 
logous sides  AB  and 
FG :  that  is 

ABODE  :  FGHIK  :  :  square  L  :  square  M. 

If  AB  were  4,  the  area  ABODE  would  be  27.5276384. 
Now,  if  i^G  were  8,  what  would  be  the  area  FGHIK? 

4'  :   8*  :  -:  27:5276384  :   110.1105536. 


48.  How  do  you  find  the  area  of  a  regular  polygon  ? 

Multiply  half  the  perimeter  of  the  figure  by  the  perpen- 
dicular let  fall  from  the  centre  on  one  of  the  sides,  and  the 
product  will  be  the  area. 


130 


BOOK    V. SECTION 


EXAMPLES. 


1.  Required  the  area  of  the  regu- 
lar pentagon  ABCDE,  each  of  whose 
sides  AB,  BC,  &c.,  is  25  feet,  and 
the  perpendicular  OP,  17.2  feet. 


We  first  multiply  one  side 
by  the  number  of  sides  and 
divide  the  product  by  2  :  this 
gives  half  the  perimeter, 
which  we  multiply  by  the 
*^    perpendicular  for  the  area. 


Operation. 


25  X  5 

2 
imeter. 
62.5  X  17.2 
the  area. 


:62.5  =:  half  the  per- 

Then, 

1075  sq.ft.  z= 


2.  The  side  of  a  regular  pentagon  is  20  yards,  and  the 

perpendicular  from  the  centre  on  one  of  the  sides  13.76382  : 

required  the  area. 

Ans.  688.191  sq.  yds. 

3.  The  side  of  a  regular  hexagon  is  14,  and  the  perpen- 
dicular from  the  centre  on  one  of  the  sides  12.1243556  : 
required  the  area. 

Ans.  509.2229352  sq.ft. 

4.  Required  the  area  of  a  regular  hexagon  whose  side 
is  14.6,  and  perpendicular  from  the  centre   12.64  feet. 

Ans.  553.632  sq.ft. 

5.  Required  the  area  of  a  heptagon  whose  side  is  19.38, 
and  perpendicular  20  feet. 

Ans.   1356.6  sq.ft. 

6.  Required  the  area  of  an  octagon  whose  side  is  9.941 
yards  and  perpendicular  12  yards. 

Ans.  477.168  sq.  yds, 

49.  The  following  table'  shows  the  areas  of  the  ten  regu- 


OF    POLYGONS. 


131 


lar  polygons  when  the  side  of  each  is  equal  to  1.     It  also 
shows  the  length  of  the  radius  of  the  inscribed  circle. 


Number  of 
sides. 

Names. 

Areas. 

Radius  of  inscribed 
circle. 

3 

Triangle, 

0.4330127 

0.2886751 

4 

Square, 

1.0000000 

0.5000000 

5 

Pentagon, 

1.7204774 

0.6881910 

6 

Hexagon, 

2.5980762 

0.8660254 

7 

Heptagon, 

3.6339124 

1.0382617 

8 

Octagon, 

4.8284271 

1.2071068 

9 

Nonagon, 

6.1818242 

1.3737387 

10 

Decagon, 

7.6942088 

1.5388418 

11 

Undecagon, 

9.3656404 

1.2028437 

'.  13 

Duodecagon, 

11.1961524 

1.8660254 

50.  How  do  you  find  the  area  of  any  polygon  from  the 
above  table  ? 

Since-  the  areas  of  similar  polygons  are  to  each  other 
as  the  squares  described  on  their  homologous  sides,  we 
have  '''.'-' 

1^  :  tabular  area  :   :  any  side  squared  :  area. 
Hence,  to  find  the  area  of  a  regular  polygon. 

1st.  Square  the  side  of  the  polygon. 

2^d.  Multiply  the  square  so  found,  by  the  tabular  area  set 
opposite  the  polygon  of  the  same  number  of  sides,  and  the 
product  will  be  the  required  area. 


•  EXAMPLES. 

r.  What  is  the  area  of  a  regular  hexagon  whose  side 
is  20? 

20^  =  400       and  tabular  area  =r  2.5980762. 
Hence, 

2.5980762  X  400  =  1039.23048  =  the  area. 
2.  What  is  the  area  of  a  pentagon  whose  side  is  25  ? 

Ans.  1075.298375. 


♦ 


132  BOOK    V. SECTION    I. 

3.  What  is  the  area  of  a  heptagon  whose  side  is  30  ? 

Ans.  3270.52116. 

4.  What  is  the  area  of  an  octagon  whose  side  is  10  feet? 

Ans.  482.84271  sq.ft. 

5.  The  side  of  a  nonagon  is  50 :  what  is  its  area  ? 

■Ans.  15454.5605. 

6.  The  side  of  an  undecagon  is  20 :   what  is  its  area  ? 

Ans.  3746.25616. 

7.  The  side  of  a  duodecagon  is  40 :   what  is  its  area  ? 

Ans.  17913.84384. 

8.  Required  the  area  of  an  octagon  whose  side  is   16. 

Ans.  1236.0773. 

9.  Required  the  area  of  a  decagon  whose  side  is  20.5. 

Ans.  3233.4912. 

10.  Required  the  area  of  a  nonagon  whose  side  is  36. 

Ans.  8011.6442. 

11.  Required  the  area  of  a  duodecagon  whose  side  is  125. 

Ans.  174939.881. 


OF    THE    CIRCLE. 

51,  How  are  the  circumferences  of  circles  to  each  other? 

The  circumferences 
of  circles  are  propor- 
tional to  their  diame- 
ters. If  we  represent 
the  diameter  AB  by  D, 
and  the  circumference 
of  the  circle  by  C,  and  the  diameter  CD  by  d,  and  the  cir- 
cumference by  c,  we  shall  haA'-e 

Did:  :    C    :    c. 


OF    THE    CIRCLE. 


133 


62.  How  many  times  greater  is  the  circumference  than  the 
diameter  of  a  aircle  1 

The  circumference  of  a  circle  is  a  little  more  than  three 
times  greater  than  the  diameter.  If  the  diameter  is  1,  the 
circumference  will  be  3.1416. 


53.   What  is  the  area  of  a  circle  equal  to  ? 

The  area  of  a  circle  is  equal  to  the 
product  of  half  the  radius,  into  the  cir- 
cumference. Thus,  the  area  of  the  cir- 
cle whose  centre  is  C,  is  equal  to  half 
the  radius  CA,  multiplied  by  the  circum- 
ference :  that  is, 

area  —  ^CA  x  circumference  ABD. 


54.  How  do  you  fnd  the  circumference  of  a  circle  when  the 
diameter  is  known  1 

Multiply  the  diameter  by  3.1416,  and  the  product  will 
be  the  circumference. 


EXAMPLES. 


1.  What  is  the  circumference  of  a  circle  whose  diame- 
ter is  17? 

We    simply   multiply    the 


number  3.1416  by  the  diam- 
eter, and  the  product  is  the 
circumference. 


Operation. 

3.1416  X  17  =  53.4072 

which  is  the  circumference. 


2.  What  is  the  circumference  of  a  circle  whose  diameter 
is  40  feet  ? 

Ans.  VZb.m^ft. 

3.  What  is  the  circumference  of  a  circle  whose  diame- 
ter is  12  feet? 

Ans.  37.6992 /^ 


134  BOOK    V. SECTION    I. 

4.  What  is  the  circumference  of  a  circle  whose  diame- 
ter is  22  yards  ? 

Ans.  69.1152  yds. 

5.  What  is  the  circumference  of  the  earth' — the  mean 
diameter  being  about  7921  miles? 

Ans.  24884.6136  miles. 

65.  Hotjp  do  you  find  the  diameter  of  a  circle  when  the  cir- 
cumference is  known  ? 

Divide  the  circumference  by  the  number  3.1416,  and  the 
quotient  will  be  the  diameter. 


EXAMPLES. 

1.  The  circumference  of  a  circle  is  69.115,2  yards:  what 
is  the  diameter  ? 

Operation. 


We  simply  divide  the  cir- 
cumference by  3.1416,  and 
the  quotient  22  is  the  diame- 
ter sought. 


3.1416)69.1152(22 
62832 
62832 
62832 


2.  What  is  the  diameter  of  a  circle  whose  circumference 
is  11652.1944  feet?    . 

Ans.  3709  ft. 

3.  What  is  the  diameter  of  a  circle  whose  circumference 
is  6850  ? 

Ans.  2180.4176. 

4.  What  is  the  diameter  of  a  circle  whose  circumference 
is  50? 

Ans.  15.915. 

5.  If  the  circumference  of  a  circle  is  25000.8528,  what 
is  the  diameter? 

Ans.  7958. 


OF    THE    CIRCLE.  135 

56.  How  do  you  find  the  length  of  a  circular  arc,  when  the 
number  of  degrees  which  it  contains,  and  the  radius  of  the 
circle  are  known  ? 

Multiply  the  number  of  degrees  by  the  decimal  .01745, 
and  the  product  arising  by  the  radius  of  the  circle. 


EXAMPLES. 

1.  What  is  the   length  of  an   arc  of  30  degrees,  in   a 
circle  whose  radius  is  9  feet? 

We    merely   multiply   the  j  Operation. 

given  decimal  by  the    num-  |   .01745  x  30  x  9  =  4.7115, 


ber  of  degrees,   and  by  the  j  which  is  the    length  of  the 
radius.  !  required  arc. 


Remark. — When  the  arc  contains  degrees  and  minutes, 
reduce  the  minutes  to  the  decimals  of  a  degree,  which  is 
done  by  dividing  them  by  60. 

2.  What  is  the  length  of  an  arc  containing  12°  10''  or 
12^°,  the  diameter  of  the  circle  being  20  yards  1 

Ans.  2.1231. 

3.  What  is  the  length  of  an  arc  of  10°  15^  or  101°,  in 
a  circle  whose  diameter  is  68  ? 

Ans.  6.0813. 

57.  How  do  you  find  the  length  of  the  arc  of  a  circle  when 
the  chord  and  radius  are  given  ? 

1st.  Find  the  chord  of  half  the  arc. 

2d.  From  eight  times  the  chord  of  half  the  arc,  subtract 
the  chord  of  the  whole  arc,  and  divide  the  remainder  by 
three,  and  the  quotient  will  be  the  length  of  the  arc, 
nearly. 


136 


BOOK    V. SECTION    I. 


EXAMPLES. 

1.  The  chord  AB  =  30  feet,  and 
the  radius  AC  ==:  20  feet:  what  is 
the  length  of  the  arc  ADB  ? 

First,  draw  CD  perpendicular  to 
the  chord  AB :  it  will  bisect  the 
chord  at  P,  and  the  arc  of  the  chord 
at  D.     Then  AP  =  15  feet.     Hence 


AC  -  AP   =  CP  :    that  is 

400  —  225  =  175,  and  VTtE  =  13.228  =  CP. 
Then,     CD  —  CP  =  20  —  13.228  =:  6.772  =  DP. 


Again,      AD  =  V  A?+  TD"  =  ^225  -f-  45.859984  : 
hence,       AD  =  16,4578  =  chord  of  the  half  arc. 
16.4578  X  8  —  30 


Then, 


33.8874  =  arc  ADB. 


2.  What. is  the  length  of  an  arc,  the  chord  of  which  is 
24  feet,  and  the  radius  of  the  circle  20  feet  ? 

Ans.  25.7309 /i. 

3.  The  chord  of  an  arc  is  16,  and  the  diameter  of  the 
circle  20  :    what  is  the  length  of  the  arc  ? 

Ans.   18.5178. 

4.  The  chord  of  an  arc  is  50,  and  the  chord  of  half 
the  arc  is  27 :   what  is  the  length  of  the  arc  ? 

Ans.  55^. 

58.  How  do  you  jind  the  area  of  a  circle  when  the  diameter 
and  circumference  are  both  known  ? 

Multiply  the  circumference  by  half  the  radius,  and  the 
product  will  be  the  area. 


OF    THE    CIRCLE.  137 

EXAMPLES. 

1.  What'  is  the  area  of  a  circle  whose  diameter  is  10, 
and  circumference  31.416? 
If  the  diameter  be  10,  the 


radius  is  5,  and  half  the  ra- 
dius 2L  :  hence  the  circumfe- 
rence multiplied  by  2l  gives 
the  area. 


Operation. 
31.416  X  21  =  78.54) 
which  is  the  area. 


2.  Find  the  area  of  a  circle  whose  diameter  is  7,  and 

circumference  21.9912  yards. 

Ans.  38.4846  yds. 

3.  How  many  square  yards  in  a  circle  whose  diameter 
is  31  feet,  and  circumference  10.9956? 

Ans.  1.069016. 

4.  What  is  the  area  of  a  circle  whose  diameter  is  100, 

and  circumference  314.16  ? 

Ans.  7854. 

5.  What  is  the  area  of  a   circle  whose   diameter  is  1, 

and  circumference  3.1416  ? 

Ans.  0.7854. 

6.  What  is  the  area  of  a  circle  whose  diameter  is  40, 

and  circumference  131.9472? 

Ans.  1319.472. 

59.  How  do  you  find  the  area  of  a  circle  when  the  diam- 
eter only  is  known  ? 

Square  the  diameter,  and  then  multiply  by  the  decimal 
.7854. 

EXAMPLES. 

1.  What  is  the  area  of  a  circle  who^e  diameter  is  5  ? 


138  BOOK    V. SECTION    I. 


We  square  the  diameter, 
which  gives  us  25,  and  we 
then  multiply  this  number 
and  the  decimal  .7854  to- 
gether. 


Operation. 

.7854 

?= 25 

39270 
15708 


area  —  19.6350 


2.  What  is  the  area  of  a  circle  whose  diameter  is  7  ? 

Ans.  38.4846. 

3.  What  is  the  area  of  a  circle  whose  diameter  is  4.5 1 

Ans,   15.90435. 

4.  What  is  the  number  of  square  yards  in  a  circle  whose 
diameter  is  IJ-  yards  ? 

Ans.   1.069016. 

5.  What  is  the  area  of  a  circle  whose  diameter  is  8.75 
feet  ? 

Ans.  60.1322  sq.ft.. 

60.  How  do  you  find  the  area  of  a  circle  when  the  cir- 
cumference  only  is  known  ? 

Multiply  the  square  of  the  circumference  by  the  decimal 
.07958,  and  the  product  will  be  the  area  very  nearly. 

EXAMPLES. 

1.  What  is  the  area  of  a  circle  whose  circumference  is 
3.1416? 


We  first  square  the  cir- 
cumference, and  then  multi- 
ply by  the  decimal  .07958. 


Operation. 


3.1416   z=  9.86965056 
.07958 


area  =  .7854  + 


2.   What  is  the   area  of  a   circle  whose   circumference 
is  91? 

Ans.  659.00198. 


OF    THE    CIRCLE.  139 

3.  Suppose  a  wheel  turns  twice  in  tracking  16^  feet,  and 
that  it  turns  just  200  times  in  going  round  a  circular  bowl- 
ing-green :   what  is  the  area  in  acres,  roods,  and  perches  ^ 

Ans.  4  A.3  R.  35.8  P. 

4.  How  many  square  feet  are  there  in  a  circle  whose 
circumference  is  10.9956  yards? 

Ans.  86.5933. 

5.  How  many  perches  are  there  in  a  circle  whose  cir- 
cumference is  7  miles  ? 

Ans.  399300.608. 

61.  Having  given  a  circle,  how  do  you  Jlnd  a  square  which 
shall  have  an  equal  area? 

1st.  The  diameter  X  .8862  =  side  of  an  equivalent 
square. 

2d.  The  circumference  x  .2821  =z  side  of  an  equivalent 
square.         ' 

EXAMPLES. 

1.  The  diameter  of  a  circle  is  100:  what  is  the  side 
of  a  square  of  an  equal  area  ? 

Ans.  88.62. 

2.  The  diameter  of  a  citcular  fish-potid  is  20  feet :  what 
would  be  the  side  of  a  square  fish-pond  of  an  equal  area  ? 

Ans.  \1.124ft. 

3.  A  man  has  a  circular  meadow,  of  which  the  diameter 
is  875  yards,  and  wishes  to  exchange  it  for  a  square  one 
of  equal  size  :   what  must  be  the  side  of  the  square  1 

Ans.  775.425. 

4.  The  circumference  of  a  circle  is  200 :  what  is  the 
side  of  a  square  of  an  equal  area  ? 

Ans.  56.42. 


140  BOOK    V. SECTION    I. 

5.  The  circumference  of  a  round  fish-pond  is  400  yards : 
what  is  the  side  of  a  square  fish-pond  of  equal  area  1 

Ans.  112.84. 

6.  The  circumference  of  a  circular  bowling-green  is  412 
yards:  what  is  the  side  of  a  square  one  of  equal  area? 

Ans.   116.2252  yds. 

62.  Having  given  the  diameter  or  circumference  of  a  circle, 
how  do  yf/u  find  the  side  of  the  inscribed  square  ? 

1st.  The  diameter  X  .7071  =  side  of  the  inscribed 
square. 

2d.  The  circumference  X  .2251  =  side  of  the  inscribed 
square . 

EXAMPLES. 

1.  The  diameter  AB  of  a  cir- 
cle is  400:  what  is  the  value  of 
AC,  the  side  of  the  inscribed 
square  ?  A 

Here 

.7071  X  400  =  282.8400  =  AC. 

2.  The  diameter  of  a  circle  is  412  feet:  what  is  the 
side  of  the  inscribed  square  ? 

Ans.  291.3252/^. 

3.  If  the  diameter  of  a  circle  be  600,  what  is  the  side 
of  the  inscribed  square  ? 

Ans.  424.26. 

4.  The  circumference  of  a  circle  is  312  feet:  what  is 
the  side  of  the  inscribed  square  ? 

Ans.  70.2312//;. 

5.  The  circumference  of  a  circle  is  819  yards:  what  is 
the  side  of  the  inscribed  square  ? 

Ans.  184.3569.^6?^. 


OF    THE    CIRCLE.  ^        141 

6.  The  circumference  of  a  circle  is  715 :  what  is  the 
side  of  the  inscribed  square  ? 

Ans.  160.9465. 

63.  How  do  you  find  the  area  of  a  circular  sector  ? 
1st.  Find  the  length  of  the  arc  by  Art.  56. 

2d.  Multiply  the  arc  by  one-half  the  radius,  and  the  pro- 
duct will  be  the  area. 

EXAMPLES. 

1.  What  is  the  area  of  the  circular 
sector  ACB,  the  arc  AB  containing  18°, 
and  the  radius  CA  being  equal  to  3  feet  ? 

First,  .01745  x  18  x  3  =  .94230  = 
length  AB. 

Then,  .94230  X  l^  =  1.41345  =  area. 

2.  What  is  the  area  of  a  sector  of  a  circle,  in  whicji 
the  radius  is  20  and  the  arc  one  of  22  degrees  ? 

Ans.  76.7800. 

3.  Required  the  area  of  a  sector  whose  radius  is  25  and 
the  arc  one  of  147°  29^ 

Ans.  804.2448. 

4.  Required  the  area  of  a  semicircle  in  which  the  ra- 
dius is  13. 

Ans.  265.4143. 

5.  What  is  the  area  of  a  circular  sector  when  the  length 
of  the  arc  is  650  feet  and  the  radius  325  ? 

Ans.   105625  sq.ft. 

64.  How  do  you  find  the  area  of  a  segment  of  a  circle  ? 

1st.  Find  the  area  of  the  sector  having  the  same  arc 
with  the  segment,  by  the  last  problem. 

2d.  Find  the  area  of  the  triangle  formed  by  the  chord 
of  the  segment  and  the  two  radii  through  its  extremities. 


142 


BOOK    v.  — SECtlON    I, 


3d.  If  the  segment  is  greater  than  the  semicircle,  add 
the  two  areas  together ;  but  if  it  is  less,  subtract  them,  and 
the  result  in  either  case  will  be  the  area  required. 


EXAMPLES. 

1.  What  is  the  area  of  the  seg- 
ment ADB,  the  chord  AB  =  24  feet, 
and  CA  =:  20  feet? 
First,       CP  =  -sT&S 


AV 


—  -^400  —  144^  16 


then, 
PZ>=  CD 


CP  =  20  -16  =  4, 
and,       AB  -  VjP^  +  p3'  =  Vl44  +  16  =  12.64911 : 

AT^T.       12.64911x8-24      ■    ^  ^^^^ 
then,         arc  ADB  =z =  25.7309. 


ADB  =  25.7309 
=  10 


arc 

half  radius 

area  sector     ADBC  =257.309 

area  CAB  =  192 


^P  =    12 

CP  =    16 

area  CAB  ==192 


65.309  =  area  of  segment  ADB. 


2.  Find  the  area  of  the  segment 
AFB,  knowing  the  following  lines, 
viz.:  AB  =  20.5;  FPz=  17.17; 
AF  =  20;  FG  =  11.5,  and  CA  = 
11.64. 


Arc    AGF  = 


FG  x8-  AF 


3  3 

sector  AGFBC  =  24  x  11.64  ==  279.36  : 

but         CP  =  FP  —  AC  —  17.17  —  11.64  =  5.53  : 
rni  '       Ar^ry        AB  X  CP        t20.5x5.53  ^  ^„^^ 

Then,  area  ACS  — =  — ^ — - — —  —  56.6825. 

'  Z  2 


OF    THE    CIRCLE. 


143 


Then,  area  of  sector  AFBC  =  279.36 

do.    of  triangle  ABC  =    56.6825 
gives  area  of  segment    AFB  =  336.0425 

3.  What  is  the  area  of  a  segment,  the  radius  of  the  cir- 
cle being  10,  and  the  chord  of  the  arc  12  yards  ? 

Ans.   16.324  sq.  yds. 

4.  Required  the  area  of  the  segment  of  a  circle  whose 
chord  is  16,  and  the  diameter  of  the  circle  20. 

Ans.  44.5903. 

5.  What  is  the  area  of  a  segment  whose  arc  is  a  quad- 
rant—the diameter  of  the. circle  being  18? 

Ans.  63.6174. 

6.  The  diameter  of  a  circle  is  100,  and  the  chord  of  the 
segment  60 :  what  is  the  area  of  the  segment  ? 

Ans.  408,  nearly. 

65.  How  do  you  find  the  area  of  a  circular  ring ;  that  is, 
the  area  included  between  the  circumferences  of  two  circles, 
having  a  common  centre  ? 

1st.  Square  the  diameter  of  each  ring,  and  subtract  the 
square  of  the  less  from  that  of  the  greater. 

2d.  Multiply  the  difference  of  the  squares  by  the  decimal 
.7854,  and  the  product  will  be  the  area. 


EXAMPLES. 

1.  In  the  concentric  circles 
having  the  common  centre  C, 
we  have 

AB=  10  yards,  and  DJS  r= 
6  yards :    what  is  the  area  of 
the     space     included    between 
them  ? 


OF  THE 


^'^^. 


UNIVERSITY  j 


144 


Then, 


BOOK    V. SECTION    I. 

IS  =  lo'  =  100 

2Ze'=    6^=    36 

Difference  =    64 

64  X  .7854  —  50.2656  =  area. 


2.  What  is  the  area  of  the  ring  when  the  diameters  of 
the  circles  are  20  and  10  ? 

Ans.  235.62. 

3.  If  the  diameters  are  20  and  15,  what  will  be  the  area 
included  between  the  circumferences  ? 

Ans,    137.445. 

4.  If  the  diameters  are  16  and  10,  what  will  be  the  area 
included  between  the  circumferences  ? 

Ans.  122.5224. 

5.  Two  diameters  are  21.75  and  9.5;  required  the  area 
of  the  circular  ring. 

Ans.  300.6609.   , 

6.  If  the  two  diameters  are  4  and  6,  what  is  the  area 
of  the  ring? 

Ans.   15.708. 

66.  How  do  you  Jind  the  area  of  an  ellipse  ? 
Multiply  the  two  axes  together,  and  their  product  by  the 
decimal  .7854,  and  the  result  will  be  the  required  area. 


EXAMPLES. 


1.  Required  the  area  of  an  ellipse, 
whose  transverse  axis  AB  =  70  feet, 
and  the  conjugate  axis  DE  =  50 
feet. 


Then, 


AB  X  DE  =  10  X  50  =  3500 : 
.7854  X  3500  =  2748.9  =  area. 


MENSURATION    OF    SOLIDS.  145 

2.  Required  the   area  of  an  ellipse  whose  axes  are  24 

and  18. 

Ans.  339.2928. 

3.  What  is  the  area  of  an  ellipse  whose  axes   are  35 
and  25  ? 

Ans.  687.225. 

4.  What  is  the   area  of  an   ellipse  whose   axes  are  80 
and  60? 

Ans.  3769.92. 

5.  What  is  the   area  of  an  ellipse  whose  axes  are  50 
and  45? 

Ans.   1767.15. 


SECTION   II. 


MENSURATION    OF    SOLIDS. 


1.  What  is  a  solid  or  body  ? 

A  solid  or  body  is  that  which  has  length,  breadth,  and 
thickness. 

2.  What  is  a  body  bounded  by  planes  called? 

Every  solid  bounded  by  planes  is  called  a  polyedron. 

3.  What  are  the  bounding  planes^  the  straight  lines,  and 
the  angular  points  called? 

The  planes  which  bound  a  polyedron  are  called  faces. 
The  straight  lines  in  which  the  faces  intersect  each  other, 
are  Called  the  edges  of  the  polyedron ;  and  the  points  at 
which  the  edges  intersect,  are  called  the  vertices  of  the 
angles,  or  vertices  of  the  polyedron.         -  ' 

7 


146 


BOOK    V. SECTION    II. 


4.   What  is  a  prism  ? 

vex  surface  ? 

A  prism  is  a  solid,  whose  ends 
are  equal  polygons,  and  whose  side 
faces  are  parallelograms. 

Thus,  the  prism  whose  lower  base 
is  the  pentagon  ABCDE,  terminates 
in  an  equal  and  parallel  pentagon 
FGHIK,  which  is  called  the  upper 
base.  The  side  faces  of  the  prism 
are  the  parallelograms  DH,  DK,  EF, 
AG,  BH.  These  are  called  the 
convex  or  lateral  surface  of  the  prism. 


What  are  its  bases?  what  its  con- 


5.    What  is  the  altitude  of  a  prism? 

The  altitude  of  a  prism  is  the  distance  between  its  upper 
and  lower  bases ;  that  is,  it  is  a  line  drawn  from  a  point 
of  the  upper  base,  perpendicular  to  the  lower  base. 


6.    What  is  a  right  prism  ? 

A  right  prism  is  one  in  which 
the  edges  AF,  EG,  EK,  HC,  and 
DI  are  perpendicular  to  the  bases. 
In  the  right  prism,  either  of  the  per- 
pendicular edges  is  equal  to  the  al- 
titude. In  the  oblique  prism  the 
altitude  is  less  than  the  edge. 


7.  How  are  prisms  distinguished  from  each  other  ? 

A  prism  whose  base  is  a.  triangle,  is  called  a  triangular 
prism  :  if  the  base  is  a  quadrangle,  it  is  called  a  quad- 
rangular prism:  if  a  pentagon,  a  pentagonal  prism  :  if  a 
hexagon,  it  is  called  a  hexagonar prism :    &c. 


MENSURATION    OF    SOLIDS. 


147 


8.    What  is  a  parallelopipedon  ?    what  a  cube  ? 

A  prism  whose  base  is  a  parallelogram,  and  all  of  whose 
faces  tire  also  parallelograms,  is  called  a 
parallelopipedon.  If  all  the  faces  are  rec- 
tangles, it  is  called  a  rectangular  paral- 
lelopipedon. If  all  the  faces  are  squares, 
it  is  called  a  cube.  The  cube  is  bounded 
by  six  equal  faces  at  right  angles  to  each 
other. 


\ 


N 


9.  How  do   the   opposite  faces  of  a  parallelopipedon  com- 
pare with  each  other?  jj g. 

The  opposite  faces  of  a  parallelopi- 
pedon are  equal  to  each  other.  Thus, 
the  parallelogram  BD  is  equal  to  the 
opposite  parallelogram  FH^  the  parallel- 
ogram BE  to  CH,  and  BG  to  AH. 

10.  What  li  the  content  of  a  solid? 

The  content  of  a  solid  is  the  number  of  cubes  which  it 
contains. 

In  order  to  find  the  content  of  a  solid,  suppose  ABCD 
to  be   the   base  of  a   parallelo- 
pipedon. 

Let  us  suppose  AB  =:  4  feet, 
and  BC  z^  3  feet.  Then  the 
number  of  square  feet  in  the 
base  will  be  equal  to  3  X  4  = 
12  square  feet.  Therefore,  12 
equal  cubes  of  one  foot  each, 
may  be  placed  by  the  side  of  each  other  on  the  base. 
If  the  parallelopipedon  be  1  foot  in  height,  it  will  contain 
12  such  cubes,  or  12  cubic  feet :  were  it  2  feet  in  height, 
it  would    contain   two   tiers   of  cubes,  or   24  cubic   feet : 


148  BOOK    V. SECTION    II. 

were  it  3  feet  in  height,  it  would  contain  three  tiers  of 
cubes,  or  36  cubic  feet.  Therefore,  the  solid  content  of  a 
parallelopipedon  is  equal  to  the  product  of  its  lengthy  breadth, 
and  height. 

11.  How  many  kinds  of  quantity  are  there  in  geometry? 
There    are  three   kinds  of  quantity  in   geometry,  viz.  : 

Lines,  Surfaces,  and  Solids ;  and  each  of  these  has  its 
own  unit. 

12.  What  are  the  units  of  these  kinds  of  quantity  ? 
The  unit  of  a  line,  which  we   have    called  the   linear 

unit,  is  a  line  of  a  known  length,  as  a  foot,  a  yard,  a 
rod,  &c. 

The  unit  of  surface  is  a  square,  whose  sides  are  the 
unit  of  length. 

The  unit  of  solidity  is  a  cube,  whose  edges  are  the  unit 
of  length. 

For  example,  if  the  bounding  lines  of  a  surface  be  es- 
timated in  yards,  the  content  will  be  square  yards  ;  and 
if  the  bounding  lines  of  a  solid  be  yards,  its  surface  will 
be  estimated  in  square  yards,  and  its  solid  content  in  cubic 
yards. 

13.  Into  how  many  parts  is  the  mensuration  of  solids  di- 
vided ? 

The  mensuration  of  solids  is  divided  into  two  parts : — 
1st.  The  mensuration  of  the  surfaces  of  solids :    and 
2dly.  The  mensuration  of  their  solidities. 

14.  How  is  a  curved  line  to  he  treated  ? 

A  curve  line  which  is  expressed  by  numbers  is  also 
referred  to  a .  unit  of  length,  and  its  numerical  value  is 
the  number  of  times  which  the  line  contains  the  unit. 

If,  then,  we  suppose  the  linear  unit  to  be  reduced  to  a 


OF    THE    PRISM. 


149 


straight  line,  and  a  square  constructed    on   this  line,  this 
square  will  be  the  unit  of  measure  for  curved  surfaces. 

15.  Repeat  the  table  of  solid  measures. 

1  cubic  foot     =1728  cubic  inches. 
1  cubic  yard    =  27  cubic  feet. 
1  cubic  rod      =  4492|  cubic  feet. 
1  ale  gallon      =r  282  cubic  inches. 
1  wine  gallon  =231  cubic  inches. 
1  bushel  =r  2150.42  cubic  inches. 

OF    THE    PRISM. 

> 

16.  How  do  you  find  the  surface  of  a  right  prism  ? 
Multiply  the  perimeter  of  the   base  by  the  altitude,  and 

the  product  will  be  the  convex  surface  :    and  to  this   add 
the  area  of  the  bases  when  the  entire  surface  is  required. 


EXAMPLES. 

1.  Find  the  entire  surfa*ce  of  the 
regular  prism,  whose  base  is  the 
regular  polygon  ABODE,  and  al- 
titude AF,  when  each  side  of  the 
base  is  20  feet,  and  the  altitude  AF 
50  feet. 


AB  +  BC+  CD-^DE  +  EA  = 
100;    and  ^F  =  50  : 
then  (AB  +  BC -\-CD-\-DE  +  EA)X  AF 


B         C 

=  convex  surface 
becomes  100  x  50  =  5000  square  feet,  which  is  the  con- 
vex surface.     For  the  area  of  the  end,  we  have 
AB    X  tabular  number  =  area  ABCDE,  (see  page  131;) 
that  is,     20*^  x  tabular  number,  or  400  X  1.720477  = 
688.1908  =:the  area  ABCDE. 


150  BOOK    V. SECTION    II. 

Then,  convex  surface  =  5000  square  feet, 

lower  base  688.1908         do. 

upper  base  688.1908         do. 

entire  surface         6376.3816         do. 

2.  What  is  the  surface  of  a  cube,  the  length  of  each 
side  being  20  feet?  ^^^^    2400  ^^ j^ 

3.  Find  the  entire  surface  of  a  triangular  prism,  whose 
base  is  an  equilateral  triangle,  having  eUch  of  its  sides 
equal  to  18  inches,  and  altitude  20  feet. 

Ans.  91.949  sq.ft. 

4.  What  is  the  convex  surface  of  a  regular  octagonal 
prism,  the  side  of. whose  base  is  15  and  altitude  12  feet? 

Ans.   1440  sq.ft. 

5.  What  must  be  paid  for  lining  a  rectangular  cistern 
with  lead  at  2c?.  a  pound,  the  thickness  of  the  lead  being 
such  as  to  require  llh.  for  each  square  foot  of  surface : 
the  inner  dimensions  of  the  cistern  being  as  follows :  viz., 
the  length  3  feet  2  inches,  the  breadth  2  feet  8  inches,  and 
the  depth  2  feet  6  inches?  ^^^    ^^  3^   ^^^ 

17.  How  do  you  find  the  solidity  of  a  prism^  parallelopi- 
pedon,  or  cube  1 

Multiply  the  area  of  the  base  by  the  perpendicular  height, 
and  the  product  will  be  the  solidity.  - 

EXAMPLES. 


1.  What  is  the  solidity  of  a  regular 
pentagonal  prism,  whose  altitude  is  20, 
and  each  side  of  the  base  15  feet? 

To  find  the  area  of  the  base  we 
have  (page  131) 


OF    THE    PRISM.  151 

15'  -  225:    and  225  x  1.7204774  =  387.107415  =  the, 
area  of  the  base  :    hence, 

387.107415  X  20  =  7742.1483  =  solidity. 

2.  What  is  the  solid  content  of  a  cube  whose  side  is  24 
^^^^®^-  .  Ans.  13824  solid  in. 

3.  How  many  cubic  feet  in  a  block  of  marble,  of  which 

the  length  is  3  feet  2  inches,  breadth  2  feet  8  inches,  and 

height  or  thickness  2  feet  6  inches  ? 

Ans.  21^  solid  ft, 

4.  How  many  gallons  of  water,  ale  measure,  will  a  cis- 
tern contain  whose  dimensions  are  the  same  as  in  the  last 
example  ? 

Ans.  129if 

5.  Required  the  solidity  of  a  triangular  prism,  whose  al- 
titude is  10  feet,  and  the  three  sides  of  its  triangular  base 
3,  4,  and  5  feet. 

Ans.  60  solid  ft. 

6.  What  is  the  solidity  of  a  square  prism,  whose  height 
is  51  feet,  and  each  side  of  the  base  li  fe^t? 

Ans.  9^  solid  ft. 

7.  What  is  the  solidity  of  a  prism,  whose  base  is  an 
equilateral  triangle,  each  side  of  which  is  4  feet,  the  height 
of  the  prism  being  10  feet? 

Ans.  69.282  solid  ft. 

8.  What  is  the  number  of  cubic  or  solid  feet  in  a  reg- 
ular pentagonal  prism,  of  which  the  altitude  is  15  feet  and 
each  side  of  the  base  3.75  feet  ? 

A?is.  362.913. 

9.  What  is   the   solidity   of  a  prism,  whose  base  is  an 

equilateral  triangle,  each  side  of  which  is  1.5  feet,  and  the 

altitude  18  feet?  .        i^Koo-m^oc      i-    j\ 

Ans.  17.53701435  cubic  ft. 


152 


BOOK    V. — SECTION    H. 


10.  What  is  the  solidity  of  a  cube,  whose   side  is   15 

inches  ? 

Ans.  1.953125  cubic  ft. 

11.  What  is  the  solidity  of  a  cube,  whose  side  is  17.5 
inches? 

Ans.  3.1015  cubic  ft. 

12.  What  is  the  solidity  of  a  prism,  whose  base  is  a 
hexagon,  each  side  of  which  is  1  foot  4  inches,  and  the 
length  of  the  prism  15  feet? 

Ans.  69.2820285  solid  ft. 

13.  What  is  the  solidity  of  a  prism,  whose  altitude  is 
30  feet,  and  whose  base  is  a  heptagon,  each  side  of  which 
is  13  feet  3  inches? 

Ans.  1913.936237175  solid  ft. 


OF    THE    PYRAMID. 


18.  What  is  a  pyramid,  and  what  are  its  parts  ? 

A  pyramid   is    a   solid,   formed   by  'S 

several  triangles  united  at  the  same 
point  iS,  and  terminating  in  the  dif- 
ferent sides  of  a  polygon  ABCDE. 

The  polygon  ABCDE,  is  called  the 
base  of  the  pyramid;  the  point  ^  is 
called  the  vertex,  and  the  triangles 
ASB,  BSC,  CSD,  DSE,  and  ESA, 
form  its  lateral,  or  convex  surface. 

19.  What  is  a  solid  angle,  and  what  is  the  least  number 
of  planes  that  can  form  one  ? 

A  solid  angle  is   the   angular   spaces  included  between 
several  planes  which   meet   at  a  point.     Thus,  the  solid 


OF    THE    PYRAMID. 


153 


angle  >S  is  formed  by  the  meeting  of  the  five  planes  ESD, 
DSC,  CSB,  BSA,  and  ASE.  The  point  S  is  called  the 
vertex  of  the  solid  angle.  Three  planes,  at  least,  are  re- 
quired to  form  a  solid  angle. 


20.  What  is  the  altitude  of  a  pyra- 
mid ? 

The  altitude  of  a  pyramid,  is  the 
perpendicular  let  fall  from  the  vertex 
upon  the  plane  of  the  base.  Thus, 
/SO  is  the  altitude  of  the  pyramid  S 
^ABCDE.  _       • 


21.  What  is  the  slant  height  of  a 
pyramid  1 

The  slant  height  of  a  regular  pyra- 
mid, is  a  line  drawn  from  the  ver- 
tex, perpendicular  to  one  of  the  sides 
of  the  polygon  which  forms  its  base. 
Thus,  SF  is  the  slant  height  of  the 
pyramid  S^ABCDE. 


22.    What  is  the  axis  of  a  pyramid  1 

^WTien  the  base  of  the  pyramid  is  a  regular 'polygon,  stnd 

the  perpendicular  SO  passes  through  the  middle  point  of 

the  base,  the  pyramid  is  called  a  regular  pyramid,  and  the 

line  SO  is  called  the  axis. 

■       7* 


154 


BOOK    V. SECTION    II. 


23.  What  is  the  frustum  of  a  pyra- 
mid? what  the  altitude  of  the  frus- 
tum? 

If  from  the  pyramid  S—ABCDE 
the  pyramid  ^S — abode  be  cut  off  by 
a  plane  parallel  to  the  base,  the  re- 
maining solid,  below  the  plane,  is 
called  the  frustum  of  a  pyramid. 

The  altitude  of  a  .frustum  is  the 
perpendicular  distance  between  the 
upper  and  lower  planes. 

24.  How  are  pyramids  distinguished  ? 

A  pyramid  whose  base  is  a  triangle,  is  called  a  tri- 
angular pyramid ;  if  the  base  is  a  quadrangle,  it  is  called 
a  quadrangular  pyramid ;  if  a  pentagon,  it  is  called  a  pen- 
tagonal pyramid ;  if  the  base  is  a  hexagon,  it.  is  called  a 
hexagonal  pyramid,  &c. 


25.    What  is  the  convex  surface  of 
a  regular  pyramid  equal  to? 

The   convex   surface   of  a  regular   - 
pyramid,  is  equal  to  the  perimeter  of 
the  base,  multiplied  by  half  the  slant 
height.     Thus,  the  convex  surface  of 
he  pyramid  S — ABODE  is  equal  to 

^SF  {AB  +  BC+CD  +  DE-\-EA.) 


26.  How  do  you  find  the  surface  of  a  regular  pyramid? 

Multiply  the  perimeter  of  the  base  by  half  the  slant 
height,  and  the  product  will  be  the  convex  surface :  to 
this  add  the  area  of  the  base,  if  the  entire  surface  is  re- 
quired. 


OF    THE    PYRAMID. 


155 


EXAMPLES. 

1.  In  the  regular  pentagonal  pyra- 
mid  S — ABODE,   the    slant   height 
SF  is  equal  to  45,  and  each  side  of 
the    base    is    15   feet :    required   the 
convex  surface,  and  also   the   entire 
surface. 
15  X  5 = 75  =i  perimeter  of  the  base, 
75  X  22i  =  1687.5  square  feet  — 
area  of  convex  surface. 

And  15'  =  225, 
then  225  X  1.7204774  z=  387.107415  = 


the  area  of  the  base. 


Hence,  convex  surface  =  1687.5 

area  of  the  base     =    387.107415 
entire  surface 


2074.607415  square  feet. 


2.  What  is  the   convex   surface   of  a   regular  triangular 

pyramid,  the  slant  height  being  20  feet,  and  each  side  of 

the  base  3  feet  1  a        an        r* 

Ans.  90  sq.ft. 

3.  What  is  the  entire  surface  of  a  regular  pyramid, 
whose  slant  height  is  15  feet,  and  the  base  a  regular  pen- 
tagon, of  which  each  side  is  25  feet  ? 

Ans.  2012.798  sq.ft. 

4.  What  is  the   entire   surface  of   a    regular   octagonal 

pyramid,  of  which  eaah  side  of  the  base  is  9.941  yards, 

and  the  slant  height  15?  .         m^o  coo  j 

,  ^  Ans.  1073.628  sq.  yds. 

27.  How  do  you  find  the  solidity  of  a  pyramid? 
Multiply  the  area  of  the  base  by  the  altitude,  and  divide 
the  product  by  three :  the  quotient  will  be  the  solidity. 


156 


BOOK    V. SECTION    II. 


EXAMPLES. 

1.  What  is  the  solidity  of  a  pyra- 
mid, the  area  of  whose  base  is  215 
square  feet,  and  the  altitude  S0=  45 
feet? 

First,       215  X  45  =  9675: 
then         9675  -^  3  =  3225 

which  is  the  solidity  expressed  in  solid 
feet. 


2.  Required  the  solidity  of  a  square  pyramid,  each  side 
of  its  base  being  30,  and  its  altitude  25. 

Ans.  7500  solid  ft. 

3.  How  many  solid  yards  are  there  in  a  triangular  pyra- 
mid, whose  altitude  is  90  feet,  and  each  side  of  its  base 

^y"^^-  ^n..  38.97114. 

4.  How  many  solid  feet  in  a  triangular  pyramid,  the  al- 
titude of  which  is  14  feet  6  inches,  and  the  three  sides 
of  its  base  5,  6,  and  7  feet?  ^^^^  ^^  ^3^^. 

('_ 

5.  What  is  the  solidity  of  a  regular  pentagonal  pyramid, 
its  altitude  being  12  feet,  and  each  side  of  its  base  2  feet? 

Ans.  27.5276  solid  ft. 

6.  How  many  solid  feet  in  a  regular  hexagonal  pyra- 
mid, whose  altitude  is  6.4  feet,  and  each  side  of  the  base 

6    inches?  .  ,    na^aA 

Ans.   1.38564. 

7.  How  many  solid  feet  are  contained  in  a  hexagonal 
pyramid,  the  height  of  which  is  45  feet,  and  each  side 
of  the  base  10  feet?  ^„..  3897.1143. 


OF  THE  FRUSTUM  OF  A  PYRAMID.         157 

8.  The  spire  of  a  church  is  an  octagonal  pyramid,  each 
side  of  the  base  being  5  feet  10  inches,  and  its  perpen- 
dicular height  45  feet.  Within  is  a  cavity,  or  hollow  part, 
each  side  of  the  base  of  which  is  4  feet  11  inches,  and 
its  perpendicular  height  41  feet :  how  many  yards  of  stone 
does  the  spire  contain?  ^^^^  32.197353. 


OF  THE  FRUSTUM  OF  A  PYRAMID. 

28.  How  do  you  Jind  the  convex  surface  of  the  frustum  of 
a  regular  pyramid  ? 

Multiply  half  the  sum  of  the  perimeters  of  the  two  bases 
by  the  slant  height  of  the  frustum,  and  the  product  will  be 
the  convex  surface. 

EXAMPLES. 

1.  In  the  frustum  of  the  regular  pen- 
tagonal pyramid  each  side  of  the  lower 
base  is  30  and  each  side  of  the  upper 
base  is  20  feet,  and  the  slant  height  fF 
is  equal  to  15  feet.  What  is  the  convex 
surface  of  the  frustum  ? 

Ans.  1875  sq.ft. 

2.  How  many  square  feet  are  there  in  the  convex  sur- 
face of  the  frustum  of  a  square  pyramid,  whose  slant  height 
is  10  feet,  each  side  of  the  lower  base  3  feet  4  inches,  and 
each  side  of  the  upper  base  2  feet  2  inches? 

Ans.  110. 

3.  What  is  the  convex  surface  of  the  frustum  of  a  hep- 
tagonal  pyramid  whose  slant  height  is  55  feet,  each  side 
of  the  lower  base  8  feet,  and  each  side  of  the  upper  base 

^^®®^'  Ans.  2310  sq.ft. 


158  BOOK    V. SECTION    II. 

29.  How  do  you  find  the  entire  surface  of  the  frustum  of 
a  regular  'pyramid! 

To  the   convex  surface,  found  as   above,  add  the  areas 

of  the  two  ends,  and  the  result  will  be  the  entire  surface. 

What  is  the  entire   surface  of  the    frustum  in  each  of 

the  last  three  examples  ? 

f  1st.  4111.62062  sq.ft. 

Ans.    )2d.   125|f  sq.ft. 

(3d.  2600.712992^5'. /^ 

30.  How  do  you  find  the  solidity  of  the  frustum  of  a 
pyramid  ? 

Add  together  the  areas  of  the  two  bases  of  the  frustum 
and  a  geometrical  mean  proportional  between  them ;  and 
then  multiply  the  sum  by  the  altitude  and  take  one-third 
of  the  product  for  the  solidity. 


EXAMPLES. 

1.  What  is  the  solidity  of  the  frus- 
tum of  a  pentagonal  pyramid,  the  area 
of  the  lower  base  being  16  and  of  the 
upper  base  9  square  feet,  the  altitude 
being  7  feet? 


First,    16  X  9  =  144 :  then  V144  =  12  the  mean. 

Then,  area  of  lower  base  z=  16 

"  upper  base  =9 

mean  of  bases  =  12 

37 

height  7 

3)259 

solidity     =  86i  solid  feet. 


OF    THE    CYLINDER.  159 

2.  What  is  the  number  of  solid  feet  in  a  piece  of  tim- 
ber whose  bases  are  squares,  each  side  of  the  lower  base 
being  15  inehes,  and  each  side  of  th&  upper  base  being  6 

inches — the  length  being  24  feet? 

Ans.   19.5. 

3.  Required  the  solidity  of  a  regular  pentagonal  frus- 
tum, whose  altitude  is  5 'feet,  each  side  of  the  lower  base 
18  inches,  and  each  side  of  the  upper  base  six  inches. 

Ans.  9.31925  solid  ft. 

4.  What  is  the  content  of  a  regular  hexagonal  frustum, 
whose  height  is  6  feet,  the  side  of  the  greater  end  18 
inches,  and  of  the  less  end  12  inches  ? 

Ans.  24.681724  cubic  ft. 

5.  How  many  cubic  feet  in  a   square  piece  of  timber, 

the  areas  of  the  two  ends  being  504  and  372  inches,  and 

its  length  311  feet? 

Ans.  95.447. 

6.  What  is  the  solidity  of  a  square  piece  of  timber,  its 
length  being  18  feet,  each  side  of  the  greater  base  18 
inches,   and  each  side  of  the  smaller  12  inches? 

Ans.  28.5  cubic  ft. 

7.  What  is  the  solidity  of  the  frUstum  of  a  regular  hex- 
agonal pyramid,  the  side  of  the  greater  end  being  3  feet, 
that  of  the  less  2  feet,  and  the  height  12  feet  ? 

Ans.  197.453776  solid  ft. 


OF    THE    CYLINDER. 

31.  What  is  a  cylinder  ?  what  its  upper ^  and  what  its 
lower  base?     What  is  the  axis? 

A  Cylinder  is  a  solid,  described  by  the  revolution  of  a 
rectangle  AEFD,  about  a  fixed  side  EF. 


160 


BOOK   V. SECTION    II. 


As  the  rectangle  AEFD  turns  around 
the  side  EF,  like  a  door  upon  its.  hinges, 
the  lines  AE  and  FD  describe  circles, 
and  the  line  AD  describes  the  convex 
surface  of  the  cylinder. 

The  circle  described  by  the  line  AE 
is  called  the  lower  base  of  the  cylinder, 
and  the  circle  described  by  DF  is  called 
the  upper  base. 

The  immoveable  line  EF  is  called  the  axis  of  the  cylin- 
der. 

A    cylinder,  therefore,   is   a   round   body   with    circular 
ends. 


32.  How  will  a  plane,  passed  through 
the  axis,  cut  the  cylinder? 

If  a  plane  be  passed  through  the  axis 
df  a  cylinder,  it  will  intersect  it  in  a 
rectangle  PG,  which  is  double  the  re- 
volving rectangle  EB. 


33.  If  a  cylinder  be  cut  by  a  plane 
parallel  to  the  base,  how  is  the  section! 

If  a  cylinder  be  cut  by  a  plane  paral- 
lel to  the  base,  the  section  will  be  a 
circle  equal  to  the  base.  Thus,  MLKN 
is  a  circle  equal  to  the  baseFGC 


OF    THE    CYLINDER. 


161 


34.  How  do  you  find  the  surface  of  a  cylinder  ? 

The  convex,  surface  of  a  cylinder  is 
equal  to  the' circumference  of  the  base, 
multiplied  by  the  altitude.  Thus,  the  con- 
vex surface  of  the  cylinder  ^C,  is  equal 
to  circumference  of  base  X  AD :  to  this 
add  the  areas  of  the  two  ends,  when  the 
entire  surface  is  required. 


EXAMPLES. 

1.  What  is  the  entire  surface  of  the 
cylinder,  in  which -4^,  the  diameter  of 
the  base,  is  12  feet,  and  the  altitude  EF 
30  feet? 

First,  to  find  the  circumference  of  the 
base,  (see  page  133,)  we  have 
3.1416  X  13  =  37.6992  r=  circumference 
of  the  base.  ' 

Then,     37.6992  x  30  =  1130.9760  =  convex  surface. 

Also,     12^  =  144 :  and  144  X  .7854  =  113.0976  =  area  of 

the  base. 

Then,  convex  surface  =  1130.9760 

lower  base  113.0976 

upper  base  113.0976 

Entire  area         =  1357.1712 


2.  What  is  the  convex  surface  of  a  cylinder,  the  diam- 
eter of  whose  base  is  20,  and  altitude  50  feet? 

Ans.  3141.6  sq.ft.    ^ 

3.  Required  the  entire  surface  of  a  cylinder,  whose  alti- 
tude is  20  feet,  and  the  diameter  of  the  base  2  feet. 

Ans.  1 31 .9472 /f. 


162 


BOOK    v.  — SECTION    II. 


4.  What  is  the  convex  surface  of  a  cylinder,  the  diam- 
eter of  whose  base  is  30  inches,  and  altitude  5  feet? 

Ans.  5654.88  sq.  inches. 

5.  Required  the  convex  surface  of  a  cylinder,  whose 
altitude  is  14  feet,  and  the  circumference  of  the  base  8 
feet  4  inches.  Ans.  116.6666,  &c.,  sq.ft. 


35.  How  do  you  find  the  solidity  of  a 
cylinder  ? 

The  solidity  of  a  cylinder, is  equal  to 
the  area  of  the  base,  multiplied  by  the 
altitude.  Thus,  the  solidity  of  the  cylin- 
der AC  is  equal  to 

area  of  base  x  FE. 


EXAMPLES. 

1.  What  is  the  solidity  of  a  cylinder, 
the  diameter  of  whose  base  is  40  feet, 
and  altitude  EF,  25  feet? 

First,  to  find  the  area  of  the  base,  we 
have,  (see  page  137,) 

40'  =  1600,  then  1600  X  .7854  = 
1256.64  =  area  of  the  base.     Then, 
1256.64  X  25  1=  31416    solid    feet, 
which  is  the  solidity. 

2.  What  is  the  solidity  of  a   cylinder,  the  diameter  of 
whose  base  is  30  feet,  and  altitude  50  feet  ? 

Ans.  35343  cuhic  ft.  ' 

3.  What  is  the   solidity  of  a   cylinder  whose  height  is 
5  feet,  and  the  diameter  of  the  end  2  feet  ? 

Ans.  15.708  solid  ft. 

4.  What  is    the    solidity  of  a    cylinder  whose  height  is 
20  feet,  and  the  circumference  of  the  base  20  feet  ? 

Ans.  636.64  cuhic  ft. 


^ OF    THE    CONE.  163 

5.  The  circumference  of  the  base  of  a  cylinder  is  20 
feet,  and  the  altitude  19.3,18  feet:  what  is  the  solidity? 

Ans.  614.93  cubic  ft. 

6.  What  is  the  solidity  of  a  cylinder  whose  altitude  is 
12  feet,  and  the  diameter  of  its  base   15  feet? 

Ans.  2120.58  cubic  ft. 

7.  Required  the  solidity  of  a  cylinder  whose  altitude  is 
20  feet,  and  the  circumference  of  whose  base  is  5  feet 
6  inches.  Ans.  48.1459  cubic  ft. 

8.  What  is  the  solidity  of  a  cylinder,  the  circumference 
of  whose  base  is  38  feet,  and  altitude  25  feet? 

Ans.  2872.838  cubic  ft. 

9.  What  is  the  solidity  of  a  cylinder,  the  circumference 
of  whose  base  is  40  feet,  and  altitude  30  feet  ? 

Ans.  3819.84  solid  ft. 

10.  The  diameter  of  the  base  of  a  cylinder  is  84  yards, 
and  the  altitude  21  feet:  how  many  solid  or  cubic  yards 
does  it  contain?  Ans.  38792.4768. 

OF    THE    CONE. 

36.  How  is  a  cone  described  ?  What  is  its  base  ?  what 
its  convex  surface?  what  its  altitude,  and  what  its  vertex? 

A  cone  is  a  solid,  described  by  the 
revolution  of  a  right-angled  triangle 
ABC,  about  one  of  its  sides   CB. 

The  circle  described  by  the  revolving 
side  AB,  is  called  the  base  of  the  cone. 

The  hypothenuse  ^C  is  called  the 
slant  height  of  the  cone,  and  the  surface 
described  by  it  is  called  the  convex  sur- 
face of  the  cone. 


164 


BOOK    V. SECTION    II. 


The  side  of  the  triangle  CB,  which  remains  fixed,  is 
called  the  axis  or  altitude  of  the  cone,  and  the  point  C 
the  vertex  of  the  cone. 


37.   What  is  the  frustum  of  a  cone  ? 

If  a  cone  be  cut  by  a  plane  paral- 
lel to  the  base,  the  section  will  be  a 
circle.  Thus,  the  section  FKHI  is  a 
circle.  If  from  the  cone  S — CDB, 
the  cone  S — FKH  be  taken  away,  the 
remaining  part  is  called  the  frustum  q 
of  a  cone. 


38.  How  do  you  find  the  surface  of  a  cone 
The    convex   surface   of  a    cone    is 

equal  to  the  circumference  of  the  base 

multiplied   by   half    the    slant    height. 

Thus,  the  convex  surface  of  the  cone 

C — AED  is  equal  to 

circumference  AED  x  \  CA  : 

to  this  add  the  area  of  the  base,  when 

the  entire  surface  is  required. 


EXAMPLES. 


1.  What  is  the  convex  surface  of 
the  cone  whose  vertex  is  C,  the  diam- 
eter AD  of  its  base  being  8^  feet,  and 
the  side  CA  50  feet? 


OF    THE    CONE. 


165 


First, 
Then, 


3.1416  X  81  =  26.7036  =  circum.  of  base. 


26.7036  X  50 


667.59  =  convex  surface. 


2.  Required  the  entire  surface  of  a  cone,  whose  side  is 
36,  and  thfe  diameter  of  its  base  18  feet. 

Ans.   1272.348  sq.ft. 

3.  The   diameter   of  the   base   is   3   feet,   and  the    slant 
height  15  feet:  what  is  the  convex  surface  of  the  cone? 

Ans.  70^686  sq.ft. 

4*  The  diameter  of  the  base  of  a  cone  is  4,5  feet,  and 
the  slant  height '20  feet:    what  is  the  entire  surface  ? 

Ans.   157.27635  sq.ft. 

5.  The   circumference   of  the  base  of  a   cone  is  10.75 
and  the  slant  height  is  18.25:  what  is  the  entire  surface? 

Ans.   107.29021  sq.ft. 


39,   How  do  you  find  the  solidity  of 
a  cone? 

The  solidity  of  a   cone  is  equal  to 
the  area  of  the  base  multiplied  by  one- 
third  of  the   altitude.     Thus,    the   so-* 
lidity  of  the  cone  C — AED  is  equal  to 

h^sQAED  X  \CB.  A 


40.  How  do  a  cone  and  cylinder,  of  the  same  base  and 
altitude,  compare  with  each  other  ? 

Since  the  solidity  of  a  cylinder  is  equal  to  the  base 
multiplied  by  the  altitude,  and  that  of  a  cone  to  the  base 
multiplied  by  one-third  of  the  altitude,  it  follows  that  if  a 
cylinder  and  cone  have  equal  bases  and  altitudes,  the  cone 
will  be  one-third  of  the  cylinder. 


J  66 


BOOK    V. SECTION    II. 


EXAMPLES. 


1 .  What  is  the  solidity  of  a  cone,  the 
area  of  whose  base  is  380  square  feet, 
and  altitude   CB  48  feet? 


We  simply  multiply  the 
area  of  the  base  by  the  al- 
titude, and  then  divide  the 
product  by  3. 


Operation. 

380 
48 
3040 
1520 


3)18240 
area  =  6080 


2.  Required  the  solidity  of  a  cone  whose  altitude  is  27 
feet,  and  the  diameter  of  the  base  10  feet. 

Ans.  706-86  cubic  ft. 

m 

3.  Required  the  solidity  of  a  cone  whose  altitude  is  10^ 
feet,  and  the  circumference  of  its  base  9  feet. 

Ans.  22.5609  cubic  ft. 

4.  What  is  the  solidity  of  a  cone,  the  diameter  of  whose 
base  is  18  inches,  and  altitude  15  feet? 

Ans.   8.83575  cubic  ft. 


5.  The  circumference  of  the  base  of  a  cone  is  40  feet, 
and  the  altitude  50  feet :   what  is-  the  solidity  ? 

Ans.  2122.1333  solid  ft. 


OF  THE  FRUSTUM  OF  A  CONE. 


167 


OF  THE  FRUSTUM  OF  A  CONE. 

41.  How  do  you  find  the  surface  of  the  frustum  of  a  cone  1 

Add  together  the  circumferences  of  the  two  bases,  and 

multiply  the   sum  by  half  the   slant  height  of  the  frustum; 

the  product  will  be  the  convex  surface,  to  which  add  the 

areas  of  the  bases,  when  the  entire  surface  is  required. 


EXAMPLES. 

,  1.  What  is  the  convex  surface  of  the 
frustum  of  a  cone,  of  which  the  slant 
height  is  Vl\  feet,  and  the  circumferences 
of  the  bases  8.4  and  6  feet  ? 


We  merely  take  the  sum 
of  the  circumferences  of  the 
bases,  and  multiply  by  half 
the  slant  height. 


Operation. 

8.4 
6 
14.4 
half  side     6.25 

area  r=  90  sq.  ft. 


2.  What  is  the  entire  surface  of  the  frustum  of  a  cone, 

the   side  being  16  feet,  and  the  radii  of  the  bases  2  and 

3  feet? 

Ans.  292.1688  sq.ft. 

3.  What  is  the  convex  surface  of  the  frustum  of  a  cone, 
the  circumference  of  the  greater  base  being  30  feet,  and 
of  the  less   10  feet;  the   slant  height  being  20  feet?       .  1 

Ans.   400  sq.  ft. 

4.  Required  the  entire  surface  of  the  frustum  of  a  cone 

whose   slant,  height  is  20  feet,   and  the   diameters  of  the 

bases  8  and  4  feet. 

'       ,  Ans.  439.824  sq.ft. 


168  BOOK    V. SECTION    II. 

5.  A  cone  whose  slant  height  is  30  feet,  and  the  circum 
ference  of  its  base  10  feet,  is  cut  by  a  plane  6  feet  from 
the  vertex,  measured  on  the  slant  height :  what  is  the  con- 
vex surface  of  the  frustum? 

Ans.  144  sq.  ft. 

42.  How  do  you  find  the  solidity  of  the  frustum  of  a  cone  ? 

1st.  Add  together  the  areas  of  the  two  ends  and  a  geo- 
metrical mean  between  them. 

2d.  Multiply  this  sum  by  one-third  of  the  altitude,  and 
the  product  will  be  the  solidity. 

EXAMPLES. 

1.  How  many  cubic  feet  in  the  frua-  ,^^^^ 

tum  of  a  cone,  whose  altitude  is  26  feet,  M 

and  the  diameters  of  the  bases  22  and         Ut 

1  ^  ^®®*  •  Mm 

First,  22'  X  .7854  =  380.134  =  area      ^W^ 
of  lower  base : 
and  18    x  .7854  =  254.47  =  area  of  upper  base. 


Then,  V380.134  x  254.47  =  311.018  =:  mean. 

26 
Then,  (380.134  +  254.47  -}-  311.018)  X  -^  =  8195.39 

which^is  the  solidity. 

2.' How  many  cubic  feet  in  a  piece  of  round  timber,  the 
diameter  of  the  greater  end  being  18  inches,  and  that  of 
the  less  9  inches,  and  the  length  14.25  feet? 

Ans.  14.68943. 

3.  What  is  the  solidity  of  the  frustum  of  a  cone,  the 
altitude  being  18,  the  diameter  of  the  lower  base  8,  and 
that  of  the  upper  base  4  ?  ^^    627.7888. 


OF    THE    SPHERE. 


169 


4.  What  is   the  solidity   of  the  frustym  o(  a   cone,   the 
altitude  being  25,  the  circumference  of  the  lower  base  20, 


and  that  of  the  upper  base   10 


Ans.  464.216. 


5.  If  a  cask,  which  is  composed  of  twd  equal  conic 
frustums  joined  together  at  their  larger  bases,  have  its  bung 
diameter  28  inches,  the  head  diameter  20  inches,  and  the 
length  40  inches,  how  many  gallons  of  wine  will  it  con- 
tain, there  being  231  cubic  inches  in  a  gallon  ?« 

Ans.  79.0613, 


OF    THE    SPHERE. 

43.   What  is  a  sphere  ?  ~ 

A  sphere  is  a  solid  terminated  by  a  curved  surface,  all 

the  points  of  which  are  equally  distant  from  a  certain  point 

within  called  the  centre. 


44.  Hvw  may  a  sphere  he  de- 
scribed 1 

The  sphere  may  be  described  by 
revolving  a  semicircle  ABD  about 
the  diameter  AT>,  The  plane  will 
describe  the  solid  sphere,  and  the 
semi-circumference  ABD  will  de- 
scribe the  surface. 


45.  Whai  is  ike  radius  of  a  sphere  ? 

The  radius  of  a  sphere  is  a  line 
drawn  from  the  centre  to  any  point 
of  the  circumference.  Thus,  CA  is 
a  radius. 


8 


170 


BOOK    V. SECTION  .11. 


46.  What  is  the  diameter  of  a 
sphere  1 

The  diameter  of  a  sphere  is  a 
line  passing  through  the  centre,  and 
terminated  by  the  circumference. 
Thus,  AD  is  a  diameter. 


47.  How  do  the  diameters  of  a  sphere  compare  loith  each 
other  ?  ■      . 

All  diameters  of  a  sphere  are  equal  to  each  other ;  and 
each  is  double  a  radius. 

48.  What  is  the  axis  of  a  spherej 

The  axis  of  a  sphere  is  any  line  about  which  it  re- 
volves; and  the  points  at  which  the  axis  meets  the^  sur- 
face, are  called  the  poles. 

49.  When  is  a  plane  said  to  oe 
tangent  to  a  sphere  1 

A  plane  ^  is  tfungent  to  a  sphere 
when  it  has  but  one  point  in  com- 
mon with  it.  Thus,  AB  is  'a  tan- 
gent plane, 

~  B  A 

50.  What  is  a  spherical  zone  T  ,  what  are  its  bases  1 
A  zone  is'  a  portion  of  the  surface 

of  a  sphere,  included  between  two 
parallel  planes  which  form  its  bases. 
Thus,  the  part  of  the  surface  inclu- 
ded between  the  planes  AF.  and  DF 
is  a  zone.  The  bases  of  this  zone 
are  two  circles  whose  diameters  are 
AE  and  DF. 


OF    THE    STHERE. 


171 


51,   When  will  a  zone  have  hut  one  base? 

One  of  the  planes  which  bound 
a  zone  may  become  tangent  to  the 
sphere,  in  which  case  the  zone  will 
havp  but  one  base.  Thus,  if  one 
plane  be  tangent  to  the  sphere  at  A, 
and'  another  plane  cut  it  in  the  cir-r 
cle  DF^  the  zone  included  between 
them  will  have  but  one  base. 


52.  What  is  a  spherical  segment? 

A  spherical  .segment  is  a  portion  of  the  solid  sphere  in- 
cluded between  two  parallel  planes.  These  parallel  planes 
are  its  bases.  If  one  of  the  planes  is  tangent  to  the  sphere, 
the  .segment  will  have  but  one  base. 

53.  What  is  the  altitude  of  a  zone  1 

The  altitude  of  a  zone  or  segment,  is  the  distance  be- 
tween the  parallel  planes  which  form  its  bases. 

54.  How  does  a  plane  cut  a.  sphere  ? 

Every  plane  passing  through  a  sphere  intersects  the  solid 
sphere  in  a  circle,  and  the  surface  of  the  sphere  in  the  cir- 
cumference of  a  circle. 


55.  When  does  a  plane  cut  a  sphere  in  a  great  circle? 
when  in  a  small  circle  ? 

If  the  intersecting  plane  passes  through  the  centre  of 
the -sphere,  the  circle  is  called  a  great  circle.  If  it  does 
not  pass  through  the  centre,  the  circle  of  section  is  called 
a  small  circle. 


172 


BOOK    V. SECTION    II. 


56.  How  do  you  Jind  the  surface 
of  a  sphere  ? 

The  surface  of  a  sphere  is  equal 
to  the  product  of  its  diameter  by 
the  circumference  of  a  great  circle. 
Thus,  the  surface  of  the  sphere 
whose  centre  is  C,  is  equal  to 
circumference  ABDE  x  AD. 


EXAMPLES. 

1.  What  is  the  surface  of  the  sphere 
whose  centre  is  C,  the  diameter  being 
7  feet? 

Ans.lb^m^^:  sq.ft. 


2.  What   is   the    surface    of   a    sphere   whose   diameter 
is  24?  An;?.   1809.5616. 

3.  Required  the   surface  of  a  sphere  whose  diameter  is 
7921  miles.  Ans.  197111024  sq.  miles. 

4.  What  is  the   surface  of  a   sphere   the  circumference 
of  .whose  great  circle  is  78.54?  Ahs.   1963.5. 

5.  What  is  the  surface  of  a   sphere  whose   diameter  is 
1|  feet?  Ans.  5.58506  sq.ft.    ' 


57.   How  do  you  find   the  solidity 
of  a  sphere? 

The  solidity  of  a  sphere  is  equal 
to  its  surface  multiplied  by  one-third 
of    the    radius.     Thus,    the     sphere 
whose  centre  is  C,  is  equal  to 
surface  x  ^  GA. 


OF  THE  SPHERE. 


173 


EXAMPLES. 

1 .  What  is  the  solidity  of  a  sphere 
whose  diameter  is  12  feet? 

First,     3.1416  X  12  =  37.6992  = 
circumference  of  sphere, 

diameter        =  12 


surface  =  452.3904 

one-third  radius  =  2 


solidity 


=z  904.7808  cubic  feet. 


2.  The,  diameter  of  a  sphere  is  7957.8:  what  is  its  so- 
^^^^^y'^'  ^  Ans.  263863122758.4778. 

3.  The   diameter  of  a  sphere  is  24   yards  :  what  is  its 

solid  content? 

Ans.  7238.2464  cubic  yds. 

4.  The  diameter  of  a  sphere  is  8 :  what  is  its  solidity  ? 

Ans.  268.0832. 

58.  What  is  a  second  method  of  finding  the  solidity  of  a 
sphere  ? 

Cube  the  diameter  and  multiply  the  number  thus  found 
by  the  decimal  .5236,  and  the  product  will  be  the  solidity. 

EXAMPLES. 


1 .  What  is  the  solidity  of  a  sphere  whose  diameter  is  20  ? 

Ans.  4188.8. 

2.  What  is  the  solidity  of  a  sphere  whose  diameter  is  6  ?  < 

Ans.  113.0976. 

3.  What  is  the  solidity  of  a  sphere  whose  diameter  is  10? 

Ans.  523.6. 


174 


BOOK    V. SECTION    II. 


OF    SPHERICAL    ZONES. 

59.  How  do  you  find  the  convex  surface  of  a  spherical 
zone  ? 

Multiply  the  height  of  the  zone  by  the  circumference  of 
a  great  circle  of  the  sphere,  and  the  product  will  be  the 
convex  surface. 

EXAMPLES. 


1 .  What  is  the  convex  surface  of 
the  zone  ABD,t\xe  height  i?j5  being 
9  inches,  and  the  diameter  of  the 
sphere  42  inches  ? 


First,     42  x  3.1416  =     131.9472  =  circumference, 
height  =  9 

surface  =  1187.5248  square  inches. 

2.  The  diameter  of  a  sphere  is  12i  feet:  M^hat  will  be 
the  surface  of  a  zone  whose  altitude  is  2  feet? 

Ans.  78.54  sq.ft. 

3.  The  diameter  of  a  sphere  is  21  inches :   what  is  the 
surface  of  a  zone  whose  height  is  A\  inches  ? 

Ans.  296.8812  sq.  in. 

4.  The  diameter  of  a  sphere  is  25  feet,  and  the  height 
of  the  zone  4  feet :  what  is  the  surface  of  the  zone  ? 

Ans.  314.16  sq.ft. 


OF    SPHERICAL    SEGMENTS. 


60.  How  do  you  find  the  solidity  of  a  spherical   "Segment 
with  one  base  ? 

1st.  To  three  times  the  square  of  the  radius  of  the  base 
add  the  square  of  the  height. 


OF    SPHERICAL    SEGMENTS. 


175 


2d.  Multiply  this  sum  lay  the  height,  and  the  product  by 
the  decimal  .523G  ;  the  result  will  be  the  solidity  of  the 
segment. 


EXAMPLES. 

1.  What  is  the  solidity  of  the  seg- 
ment ABD,  the  height  BE  being 
4  feet,  and  the  diameter  AD  of  the 
base  being  14  feet? 

First, 

(7^X  3  +7)  =  147+  16=  163. 

Then,  163  X  4  x  .5236  =  341,3872  solid  feet,  which  is 
the  solidity  of  the  segment. 

2.  What  is  the  solidity  of  the  segment  of  a  sphere, 
whose  height  is  4,  and  the  radius  of  its  base  8  ? 

Ans.  435.6352. 

3.  What  is  the  solidity  of  a  spherical  segment,  the  di- 
ameter of  its  base  being  17.23368,  and  its  height  4.5? 

Ans.  572.5566. 

4.  What  is  the  solidity  of  a  spherical  segment,  the  di- 
ameter of  the  sphere  being  8,  and  the  height  of  the  seg- 
ment 2  feet? 

Ans.  41.888  cubic  ft. 

5.  What  is  the  solidity  of  a  segment,  when  the  diameter 
of  the  sphere  is  20,  and  the  altitijde  of  the  segment  9  feet  ? 

Ans.  1781.2872  cubic  ft. 

61.  How  do  you  find  the  solidity  of  a  spherical  segment 
having  two  bases  1 

To  the  sum  of  the  squares  of  the  radii  of  the  two  bases 
add  one-third  of  the  square  of  the  distance  between  them ; 
then  multiply  this  sum  by  the  breadth,  and  the  product  by 
1.5708,  and  -the  result  will  be  the  solidity. 


176  BOOK    v.-    -SECTION    II. 

EXAMPLES. 

i.  What  is  the  solid  content  of  the 
zone  ADFE,  the  diameter  of  whose  ^ 
greater  base  DF  is  equal  to  30  inches, 
and  the  less  diameter  AE  15  inches, 
and  the  distance  between  the  two  ba- 
ses 10  inches? 

Now,  rby  the  rule 

[(lOf +(7.5)»  +  ^^']  X  10  X  1.5708 

=1  (100  +  56.25  4-  33.33)  x  10  X  1.5708 

=  189.58  X  10  X  1.5708  =:  2977.92264  solid  inches. - 

2.  What  is  the  solid  content  of  a  zone,  the  diameter  of 
whose  greater  base  is  24  inches,  the  less  diameter  20 
inches,  and  the  distance  between  the  bases  4  inches  1 

Ans.   1566.6112  solid  in. 

3.  What  is  the  solidity  of  the  middle  zone  of  a  sphere, 
the  diameter  of  whose  bases  are  each  3  feet,  and  the  dis- 
tance between  them  4  feet?  Ans.  Ql.lSAQ  solid^ft, 

or    THE    SPHEROID. 

62.  What  is  a  spheroid  ? 

A  spheroid  is  a  solid,  described  by  the  revolution  of  an 
ellipse  about  either  of  its  axes. 

63.  What  is  the  difference  between  a  prolate  and  an  oblate 
spheroid  ? 

If  an  ellipse  ACBD  be  re- 
volved about  the  transverse  or 
longer  axis  AB,  the  solid  de- 
scribed is  called  a  prolate 
spheroid ;  and  if  it  be  revolved 
about   the   shorter  axis    CD,  c 

the  solid  described  is  called  an  oblate  spheroid. 


OF    THE    SPHEROID.  177 

64.  What  is  the  form  of  the  earth? 

The  earth  is  an  oblate '  spheroid,  the  axis  about  which 
it  revolves  being  about  34  miles  shorter  than  the  diameter 
perpendicular  to  it.  - 

65.  How,  do  you  find  the  solidity  of  an  ellipsoid?        ':S 
Multiply  the  fixed  axis  by  the   square  of  the   revolving 

axis,   and  the   product   by  the   decimal   .5236;    the   result 
will  be  the  required  solidity. 

EXAMPLES. 

1.  In  the  prolate  spheroid 
ACBD,  the  transverse  axis 
AB  =  90,  and  the  revolving 
axis  CD  =:  70  feet :  what  is 
the  solidity  ?  ^""^^^B 

Here,    AB  =  90  feet:    'cff  -  70'  =  4900  :  hence 
AB  X  Cff  X  .5236  =  90  X  4900  x  .5236  1=  230907.6 
cubic  feet,  which  is  the  solidity. 

2.  What  is  the  solidity  of  a  prolate  spheroid,  whose 
fixed  axis  is  100,  and  revolving  axis  6  feet? 

/     ^  Ans.  1884:96, 

3.  What  is  the  solidity  of  an  oblate  spheroid,  whose  fixed 

axis  is  60,  and  revolving  axis  100?, 

Ans.  314160. 

4.  What  is  the  solidity  of  a  prolate  spheroid,  whose  axes 
are  40  and  50?  Ans.  41888. 

5.  What  is  the  solidity  of  an  oblate  spheroid,  whose  axes 
are  20  and  10?    .      ,  Ans.  2094.4. 

6.  What  is  the  solidity  of  a  prolate  spheroid,  whose  axes 
are  55  and  33?  Ans.  31361.022. 

8* 


178 


BOOK    V. — SECTION    II. 


OF    CYLINDRICAL    RINGS. 

'  66.  How  is  a  cylindrical  ring  formed  ? 

A  cylindrical  ring  is  formed  by- 
bending  a  cylinder  until  the  two 
ends  meet  each  other.  Thus,  if  a 
cylinder  be  bent  round  until  the 
axis  takes  the  position  mon,  a  solid 
will  be  formed,  which  is  called  a 
cylindrical  ring. 

The  line  AB  is  called  the  outer,  and  cd  the  inner  di- 
ameter. 

67.  How  do  you  find  the  convex  surface  of  a  cylindrical 
ring  ? 

1st.  To  the  thickness  of  the  ring  add  the  inner  diameter. 

2d.  Multiply  this  sum  by  the  thickness,  and  the  product 
by  9.8696 ;  the  result  will  be  the  airea. 


EXAMPLES. 

-1.  The  thickness  Ac  of  a  cylin- 
drical ring  is  3  inches,  and  the  i^n- 
ner  diameter  cd  is  12  inches:  what 
is  the  convex  surface  ? 

Ac -{-  cd  =z  ^ -{-  12  —  15  :  then 
15  X  3  X  9.8696  =  444.132  square 
inches  =  the  surface. 

2.  The  thickness  of  a  cylindrical  ring  is  4  inches,  and 
the  inner  diameter  18  inches:  what  is  the  convex  surface? 

-471^.  868.52  sq.  in. 

3.  The  thickness  of  a  cylindrical  ring  is  2  inches,  and 
the  inner  diameter  18  inches :  what  is  the  convex  surface  ? 

Ans.  394.784  sq.  in. 


OF    THE    FIVE    REGULAR    SOLIDS.  179 

68.  How  do  you  find  the  solidity  of  a  cylindrical  ring  1 

1st.  To  the  thickness  of  the  ring,  add  the  inner  diam- 
eter. •    ' 

2d.  Multiply  this  sum  by  the  square  of  half  the  thick- 
ness, and  the  product  by  9.8696  ;  the  result  will  be  the 
required  solidity. 

EXAMPLES. 

1 .  What  is  the  solidity  of  an  anchor-ring,  whose  inner 
diamieter  is  8  inches,  and  thickness  in  metal  3  inches? 

8  +  3  =  11  :  then,  11  X  (f  )2  X  9.8696  =  244.272|,  which 
expresses  the  solidity  in  cubic  inches. 

2.  The  inner  diameter  of  a  cylindrical  ring  is  18  inches, 
and  the   thickness   4  inches :    what  is   the  solidjty  of  the 

^^^^  •  ^  '  Ans.  868.5248  cuhic  in. 

3.  Required  the  solidity  of  a  cylindrical  ring,  whose 
thickness  is  2  inches,  and  inner  diameter  12  inches  ? 

Ans.   138.1744  cuhic  in. 

4.  What  is  the  solidity  of  a  cylindrical  ring,  whose 
thickness  is  4  inches,  and  inner  diameter  16  inches? 

Ans.  789.568  cubic  in. 


OF    THE    FIVE    REGULAR    SOLIDS. 

.  6&.  A  regular  solid  is  one  whose  faces  are  all  equal  poly- 
gons, and  whose  solid  angles  are  equal.  There  are  five 
such  solids.. 


180 


BOOK    v.— SECTION    ll. 


1.    The    tetraedron^   or    equilateral   pyramid,    is    a    solid 
bounded  by  four  equal  triangles. 


Pyramid  unfolded. 


Pyramid. 


2.  The   hexaedron^  or  cube,   is   a   solid  bounded  by  six 
equal  squares. 


c 


■ 


Cube  unfolded. 


Cube. 


3.  The  octaedron,  is  a  solid  bounded  by  eight  equal  tri- 
angles. 


Octaedron  unfolded. 


Octaedron. 


OP    THE    FIVE    REGULAR    SOLIDS. 


181 


4.  The  dodecaedron  is  a  solid  bounded  by  twelve  equal 
pentagons. 


Dodecaedron  uiifoldf  d. 


L'o   ccaeoro!).' 


5.  The   icosaedron   is  a  solid  bounded  by  twenty  equal 
triangles. 


Icosaedrx>n  unfolded. 


Icosaedror 


6.  The  regular  solids  may  easily  be  made  of  paste- 
board. 

Draw  the  figures  of  the  unfolded  regular  solids  accurately-oii 
pasteboard,  and  then  cut  through  the  bounding  line? :  this 
will  give  figures  of  pasteboard  similar  to  the  diagrams. 
Then,  cut  the  other  lines  half  through  the  pasteboard  ;  after 
which,  fold  up  the  parts,  and  glue  them  together,  and  you 
will  form  the  bodies  which  have  been  described. 


182 


BOOK^  V. SECTION    II. 


The   following  table   shows  the  surface   and  solidity  of 
each  of  the  regular  solids,  when  the  linear  edge  is  unity. 


No.  of  sides. 

Names. 

Surfaces, 

Solidities. 

4 

Tetraedon 

1.73205 

0.117B5 

6             Hexaedron 

6.00000 

1.00000 

8 

Octaedron 

3.46410 

0.47140 

12 

Dodecaedron 

20.64578 

7.66312 

20 

Icosaedron 

8.66025 

2.18169 

69.  How  will  you  find  the  surface  of  a  regular  solid,  token 
the  length  of  the  linear  edge  is  given  ? 

Multiply  the  square  of  the  linear  edge  by  the  tabular 
number  in  the  column  of  surfaces,  and  the  product  will  be 
the  surface  required. 

EXAMPLES. 

1.  The  linear  edge  of  a  tetraedron  is  3:  what  is  its 
surface?  , 

The  tabular  area  is  1.73205.     Then, 

3^  =  9 ;  and  1.73205  X  9  ==  15.58845  =  surface. 

2.  The  linear  edge  of  an  octaedron  is  5 :  what  is  its 
surface  ? 

The  tabular  area  is  3.46410.     Then, 

?  =  25  ;  and  3.46410  X  25  =r  86.6025    =  surface. 

3.  The  linear  edge  of  an  icosaedron  is  6 :  what  is  its 
surface  ? 

The  tabular  area  is  8.66025.     Then, 

6*  =  36 ;  and  8.66025  x  36  =  311.769  =  surface. 


m  OF    THE    FIVE    REGULAR    SOLIDS.  183 

70.  How  do  you  find  the  solidity  of  a  regular  solid,  when 
the  length  of  a  linear  edge  is  known? 

Multiply  the  cube  of  tke  linear  edge  by  the  tabular  num- 
ber in  the  column  of  solidities,  and  the  product  will  be  the 
solidity  required. 

EXAMPLES.  '.;._ 

1.  What  is  the  solidity  of  a  regular  tetraedron  whose 
side  is  6  ? 

The  tabular  number  in  the  column  of  solidities  is  0.11785. 
Then, 

''    ?:=  216;  and  0.11785  X  216  =  25.4556. 

2.  What  is  the  solidity  of  a  regular  octaedron  whose 
linear  edge  is,  8? 

The  tabular  number  in  the  column  of  solidities  is  0.47140. 
then, 

8^  =  512;  and  0.47140  X  512  =  241.35680  =  solidity. 

3.  What  is  the  solidity  of  a  regular  dodecaedron  whose 
linear  edge  is  3  ? 

The  tabular  number  in  the  column  of  solidities  is  7.66312. 
Then, 

3^  =  27 ;  and  7.66312  X  27  =  206.90424  —  solidity-.  ^ 

4.  What  is  the  solidity  of  a  regular  icosa.edron  whgse, 
linear  edge  is  3  ? 

The  tabular  number  in  the  column  of  solidities  is  2.18169. 
Then, 

3*  =  27 ;  and  2.18169  x  27  =  58.90563  =  soKdity.       ' 


184  BOOK   VI. — SECTION    I. 


BOOK  VI. 

ARTIFICERS'    WORK. 

SECTION   I.  -^ 

OF    MEASURES. 

1.  What  is  the  carpenter's  r:ule  used  for? 

The  carpenter's  rule,  sometimes  called  the  sliding  rule, 
is  used  for  the  measurement  of  timber,  and  artificers'  work. 
By  it  the  dimensions  are  taken,  and  by  mea^s  of  certain 
scales,  the  superficial  and  solid  contents  may  be  computed. 

2.  Describe  the  rule . 

The  rule  consists  of  two  equal  pieces  of  box  wood,  each 
one  foot  long,  and  connected  together  by  a  folding  joint. 

One  face  of  the  rule  is  divided  into  inches,  h^lf  inches, 
quarter  inches,  eighths  of  inches,  and  sixteenths  of  inches 
When  the  rule  is  opened,  the  inches  are  numbered  from  1 
to  23,  the  last  number   24,  at  the  end,  being  omitted. 

3.  How  is  the  edge  of  the  rule  divided? 

The  edge  of  the  rule  is  divided  decimally ;  that  is,  each 
foot  is  divided  into  ten  equal  parts,  and  each  of  those  agaiji 
into  ten  parts,  so  that  the  divisions  on  the  edge  of  the  scale 
are  hundredths  of  a  foot.  The  hundredths 'are  numbered 
on  each  arm  of  the  scale,  from  the  right  to  the  left. 


OF    MEASURES.  185 

4.  Ho\D  are  inches  cJianged  to  the  decimal  of  a  foot  ?    . 
By  means  of  the  decimal  divisions  it  is  easy  to  convert 

inches  into  the  decimal  of  a  foot. 

Thus,  if  we  have  6  inches,  we  find  its  corresponding: 
decimal  on  the  edge  of  the  rule  to  be  50  hundredths  of  a 
foot,  or  .50.  Also  9  inches  correspond  to  .75  ;  8  inches  t( 
.67  nearly,  and  3  inches  to  .25. 

5.  How  are  feet  and  inches  multiplied  hy  means  of  deep 
mals  ? 

The  multiplication  of  numbers  is  more  easily  made  when 
the  numbers  are  expressed  decimally  than  when  expressed 
in  feet  and  inches. 

Let  us  take  an  example.  A  board  is  12  feet  6  inches 
long,  and  2  feet  3  inches  wide :  how  many  square  feet 
does  it  contain? 

We  see  from  the  edge  of  the  rule,  that  6  inches  cor- 
respond to  .50,  and  3  inches  to  .25.     Hence,  we  have 

By  cross  multiplication.  By  decimals.    • 

12  6^  12.50 

2  3^  2.25 
25                                                              6250 

3  1^  6^''  ■  •    .  2500 

28  V  6''  content.  ^500         _ 


28.1250  content. 


6.    What  are  the  objects  of  the  scale  marked  M  and  E  ? 

Besides  the  scale  of  feet  and  inches,  already  referred 
to,  there  are,  on  the  same  side,  two  small  scales,  marked 
M  and  E  ;  the  first  is  numbered  from  1  to  36,  and  the  sec- 
ond from  1  to  26.  The  object  of  these  scales  is  to  change 
a  square  into  what  is  called  in  carpentry  an  eight  square^ 
or  regular  octagon. 


186  BOOK    VI. SECTION    I. 

7.  Explain  the  use  of  the  one  marked  M. 

Having  formed  the  square  which  is  to  be  changed  to 
the  octagon,  find  the  middle  of  each  side,  and  then  the 
divisions  of  the  scale  marked  M  show  the  distances  to  be 
laid  off  on  each  side  of  the  centre  points,  to  give  the  an- 
gles of  the  octagon. 

For  example,  if  the  side  of  the  square  is  6  inches,  the 
distance  to  be  laid  off  is  found  by  ^extending  the  dividers 
from  1  to  6.  If  the  side  of  the  square  is  12  inches,  the 
distance  to  be  taken  reaches  from  1  -to  12  ;  and  so  on  for 
any  distance  from  1  to  36. 

8.  Explain  the  use  of  the  one  marked  E. 

The  scale  marked  E  is  for  the  same  object,  only  the 
distances  are  laid  off  from  the  angular  points  of  the  square 
instead  of  from  the  centre. 

Thus,  if  we  have  a  square  whose  side  is  9  inches,  and 
wish  to  change  it  into  an  octagon,  take  from  the  scale  E 
the  distance  from  1  to  9,  and  mark  it  off  from  each  angje 
of  the  square,  on  the  sides  :  then  join  the  points,  and  the 
figure  so  formed  will  be  a  regular  octagon. 

If  the  side  of  the  square  is  18  inches,  the  distance  to 
be  taken  reaches  from  I  to  18,  and  so  fbr  any  distance 
between  1  and  26,  the  numbers  on  the  scale  pointing  out 
the  distances  to  be  laid  off  when  the  side  of  the  square  is 
expressed  in  inches.  * 

9.  What  scales  are  on  the  opposite  face  of  the  rule,  and 
how  are  they  designated? 

Turning  the  rule  directly  over,  there  will  be  seen  on 
one  arm  several  scales  of  equal  parts,  which  are  similar 
to  those  described  at  page  36.. 

Fitting  into  the  other  arm  is  a  small  brass  slide,  of  the 
same  length  as  the  rule.     On  the  face  of  the  slide  are  two 


OF    MEASURES.  187 

ranges  of  divisions,  which  are  precisely  alike.  The  upper 
is  designated  by  the  letter  B,  and  is  to  be  used  with  the 
scale  on  the  rule  directly  above,  which  is  designated  by  A  ; 
tha  lower  divisions  on  the  slide  designated  by  the  letter  C, 
are  to  be  used  with  the  scale  mark  girt  line,  and  also 
designated  by  the  letter  D.  The  scales  B  and  C  on  the 
slide,  are  .  numbered  1,  2,  3,  4,  5,  6,  7,  8,  9,  and  1,  from 
the  left  hand  towards  the  right.  From  the  middle  point 
1,  the  number^  go  on,  12,  2,-3,  4,  5,  6,  7,  8,  9,  and  10. 
Now,  the  values  which  the  parts  of  this  scale  may  repre- 
sent will  depend  on  the  value  given  to  the  unit  at  the  left 
hand.  If  the  unit  at  the  left  be  called  1,  then  the  1  at  the 
centre  point  will  represent  10,  and  the  2  at  the  right  20, 
the  3  at  the  right  30,  and  the  ten  100,.  and  similarly  for 
the  intermediate  divisions. 

If  the  left-hand  unit  be  called  10,  then  the  1  at  the 
centre  point  will  represent  one  hundred ;  the  2,  two  hun- 
dred ;  the  3,  three  hundred ;  and  so  on  for  the  divisions 
to  the  right.  "  , 

10.  How  do  you  multiply  two  numbers  together  by  the  sli' 
ding  rule  ?   . 

1st.  Mark  a  number  on  the  scale  A  to  represent  the 
multiplier. 

2d.  Then  shove  the  slide  until  1  on  B  stands  opposite 
the  multiplier  on  A. 

3d.  Then  pass  along  on  5  until  you  find  a  number  to 
represent  the  multiplicand ;  the  number  opposite  on  A  will 
represent  the  product. 

EXAMPLES. 

1.  Multiply  24  by  14.  ,  ^ 

Move  the  slide  until  1  on  i?  is  opposite  the  second  long 
mark  at  the  right  of  12,  which  is  the  division  correspond- 


188  BOOK   VI. SECTION    I. 

ing  to  i4.  Then  pass  along  B  to  the  fourth  of  the  larger 
lines  on  the  right  of  2  :  this  line  marks  the  division  on 
the  scale  A,  which  shows  the  product.  Now  we  must  re- 
mark that  the  unit  on  the  product  line  is  always  ten  times 
greater  than  the  unit  1  at  the  left  of  the  slide :  and  since 
in  the  example  this  unit  was  10,  it  follows  that  the  3  on 
A  will  stand  for  300,  and  each  of  the  smaller  divisions  for 
10;  hence  the  product  as  shown  by  the  scale  is  nearly 
340,  and  by  judging  by  the  eye,  we  write  it  336. 

2.  What  is  the  product  of  36  by  22  ? 

Move  the  slide  till  1  on  5  stands  at  22  on  A ;  then 
pass  along  on  B  to  the  6th  line  between  3  and  4 :  the 
figures  oh  A  will  then  stand  for  hundreds,  and  the-  pro- 
duct will  be  pointed  out  a  little  to  the  right  of  the  9th 
line,  between  7  and  8;    or  it  will  be  792. 

3.  A  board  is  16  feet  9  inches  long,  and  15  inches,  or 
1  foot  and  3  inches  wide  :  how  many  square  feet  does  it 
contain  ? 

First,  16  feet  and  9  inches  =  16.75  feet; 
and  «15  inches  =  1.25  feet: 
Place  1  on  5  at  the  line  corresponding  to  16,  between 
12  and  2  on  ^,  and  then  move  over  three-fourths  of  the 
distance  to  the  next  long  line  to  the  right.  Then  looking 
along  on  A,  one  quarter  of  the  distance  between  1  and  2, 
we  find  the  area  of  the  board -to  be  21  feet,  which  is  cor- 
rect, very  nearly. 

4.  The  length  of  a  board  is  15  feet  8  inches,  and  the 
breadth  1  foot  6  inches;    what  is  the  superficial  content? 

15  feet  8  inches  ■=:  15.7  nearly. 

1  foot  6  inches  =z    1.5  feet. 

Then,  place   1   on  B  at  15.7  on  A^  and  1   and  a  half  on 

B  will  mark  23  and  a  half  feet  on  A^  which  is  the  area 

very  nearly. 


OF    MEASURES.  189 

1 1.  Explain  the  manner  m  which  the  girt-Une  is  num- 
bered. 

Below  the  slide,  and  on  the  same  sidife  with  the  scales 
already  described,  is  a  row  of  divisions  marked  girt-line, 
and  numbered  from  4  to  40.  This  line-  is  also  designated 
en  the  scale  by  the  letter  D.  TJie- object  of  this  girt-line, 
which  is  to  be  used  in  conjunction  with  the  sliding  saale, 
is  to  find  the  solid  content  of  timber. 

12.  What  is  the  quarter-girt,  and  how  do  you  find  it? 

The  quarter-girt,  as  it  is  called  in  the  language  of  me- 
chanics, is  one  quarter  the  circumference  of  a  stick  of 
timber  at  its  middle  point.  The  quarter-girt,  in  squared 
timber,  is  found  by  taking  a  mean  between  the  breadth  and 
thickness.'  ":    - 

Thus,  if  the  breadth  at  the  middle  point  is  4  feet  6  inches, 
and  the  thickness  3  feet  4  inches,  we  have 

^ft.    in. 

4     6  breadth 

3     4  depth  ,  ' 

2)7  10  .  ,       " 

3   11  quarter-girt. 


and  hence  the  quarter-girt  is  3  feet  11  inches.       .    - 

13.  When  a  stick  of  timber  tapers  regularly,  how  do  you 
find  the  quarter-girt?  ' 

If  a  stick  of  timber  tapers  regularly  from  one  end  to  the 
other,  the  breadth  and  depth  at  the  middle  point  may  be 
found  by  taking  the  mean  of  the  breadth  and  depth  at  the 
ends.  ' 

Thus,  if  the   breadths  at  the  ends  are  1   foot  6  inches, 
and  1  foot  3  inches,  the  mean  breadth  will  be  1  foot  4^, 
inches.     And,  if  the  depths  at  the  ends  are  1  foot  3  inches,^ 


190  BOOK    VI. SECTION    I. 

and  1  foot,  the  mean  depth  or  thickness  will  be  1  foot  1^ 
inches ;   and  the  quarter-girt  willbe  1  foot  3  inches. 

14.  How  do  you  jind,  hy  the  sliding  rule,  the  solid  con- 
tent of  a  stick  of  timber,  when  the  length  and  quarter-girt 
are  known  ? 

1st.  Reduce  the  length  of  the  timber  to  feet  and  deci- 
mals ^of  a  foot,  and  the  quarter-girt  to  inches. 

'2d.  Note  on  scale  C  the  number  whi<;h  expresses  the 
length,  and  move  the  slide  until  this  number  falls  at  12  on 
the  girt-line. 

^  3d.  Pass  along  on  the  girt-line  till  you  find  the  number 
which  expresses  the  quarter-girt  in  inches,  and  the  division 
which  it  marks  on  C  will  shoAy  the  content  of  the  timber 
in  cubic  feet. 

EXAMPLES. 

1.  A  piece  of  square  timber  is  3  feet  9  inches  broad, 
2  feet  7  inches  thick,  and  20  feet  long:  how  many  solid 
feet  does  it  contain? 

ft.  in. 
3    9. 
2    7 


2)6    4 


3    2  quarter-girt  =  38  inches. 


Now,  move  the  slide  until  20  on  C  falls  at  12  on  the 
girt-line.  If  we  take  1  on  C  at  the  left  for  10,  2  will 
represent  20,  which  is  placed  opposite  12  on  D.  Then 
passing  along  the  girt-line  to  division  38,  We  find  the  con- 
tent on   C  to  be  a  little  over  200,  say  2001. 

2.  The  length  of  a  piece  of  timber  is  18  feet  6  inches, 
the  breadths  at  the  greater  and  less  ends  are  1  foot  6 
inches,   and    1    foot   3   inches ;    and  the   thickness    at   the 


OF    MEASURES.  191 

greater  and  les^  end,  1  foot  3  inches  and  1  foot :  what  is 
the  solid  content  ? 

Here,  the  mean  ■  breadth  is  1  foot  4^  inches,  the  mean 
thickness  1  foot  H  inches,  and  the  quarter-girt  1  foot  3 
inches,  or  15  inches. 

Therefore,  place  18.5  on  C,  at  12, on  D,  and  pass  along 
the  girt-line  to  15;  the  number  on  C,  which  is  a  little 
more  than  28  and  a  half,  will  express  the  solid  content. 

TABLE  FOR  BOARD  MEASURE. 

15.  Explain  the  table  rule  for  finding  the  content  of  boards. 

Besides  the  carpenter's  rule  with  a  slide,  which  we  have 
just  described,  there  is  another  folding  rule  without  a  slide, 
and  on  the  face  of  which  is  a  table  to  show  the  content 
of  a  board  from  1  to  20  feet  in  length,  and  from  6  to  20 
inches  in  width. 

The  upper  line  of  the  table  shows  the  -length  of  the 
board  in  feet,  and  the  column  at  the  left  shows  the  width 
of  the  board  in  inches,  from  6  to  20.  For  convenience, 
however,  the  -table  is  often  divided  into  two  parts,  which 
are  placed  by  the  side  of  each  X)ther. 

EXAMPLES.  •  ' 

1.  If  your  board  is  6  inches  wide,  and  14  feet  long,  cast 
your  eye  along  the  top  line  till  you  come  to  14  ;  directly 
under  you  will  find  7,  which  shows  that  the ,  board  contains 
7  square  feet. 

2.  If.  your  board  is  10  inches  wide,  and  16  feet  long, 
cast  your  eye  along  the  top  line  till  you  come  to  16  ;  then 
pass  along  down  till  you  come  to  the  line  of  10 :  the  num- 
ber thus  found  is  13-4,  which  shows  that  the  board  con- 
tains 13  and  4  twelfths  square  feet. 


192  BOOK    VI. SECTION    I. 

The  right-hand  side  of  the  table  begins  at  13  inches  on 
the  left-hand  column. 

3.  What,  ig  the  content  of  a  board  which  is  13  feet  long, 
and  1 9  inches  wide  ? 

Look  along 'the  upper  line  to  13;  then  descend  to  the 
line  19,  where  you  will  find  the  number  20-7,  which  shows, 
that  the  board  contains  20  and  7  twelfths  square  feet. 

4.  If  your  board  is  17  inches  wide,  and  14  feet  long, 
you  will  look  under  14  till  you  come  on  to  the  line  17, 
where  you  will  find  the  number  19-10;  which  shows  that 
the  board  contains  19  and  10  twelfths  square  feet. 

5.  If  you  have  a  board  24  feet  long,  and  20  inches  wide, 
first  take  the  area  for  20  feet  in  length,  and  then  for  4 
feet.     Thus, 

for  20  feet  by  20  inches,     33  4 
for  4  feet  by  20  inches,         6  8 

their  sum  gives     40  0  square  feet. 

Note. — Add  as  above  for  any  different  lengths  or  widths. 
,  If  your  stuff  is  1^  inches  thick,  add  half  to  it. 
If  2  inches  thick,  you  must  double  it. 
The  table  on  the  four-fold  Rule  is  not  divided. 

BOARD    MEASURE. 

16.  Explain  the  hoard  measure,  and  the  manner  of  using  it. 

This  is  a  measure  two  feet  in  length,  of  an  octagonal 
form,  that  is,  having  eight  faces.    ' 

On  the  line  running  round  the  measure,  at  the  centre, 
we  find  the  faces  of  the  measure  marked,  in  succession, 
by  the  figures  8,  9,  10,  11,  12,  13,  14,  and  15;  and  we 
shall  designate  each  face  by  the  figure  which  thus  marks 
it.     We  will  likewise  observe,  that  figures   corresponding 


OF    TIMBER    MEASURE.  193 

to  these,   are    also   sometimes   placed   at   one   end  of  the 
measure. 

Now,  these  figures  at  the  centre  of  the  measure  corre- 
spond to  the  length  of  the  board  to  be  measured.  Thus, 
if  the  board  were  13  feet  in  length,  place  the  thumb  on 
the  line  13  at  the  centre,  and  then  apply  the  measure 
across  the  board,  and  the  number  on  the  face  13,  which 
the  width  of  the  board  marks,  will  express  the  number  of 
square  feet  in  the  board.  Thus,  if  the  width  of  the  board 
extended  from  1  to  15,  the  board  would  contain  15  square 
feet. 

If  the  board  to  be  measured  was  14  feet  long,  its  con- 
tent would  be  measured  on  face  14..  If  the  board  were 
18  feet  long,  measure  its  width  on  face  8,  and  also  on 
face  10,  and  take  the  sura  for  the  true  content  of  the 
board. 
The  Measured  described  above,  are  made  by  Jones  &  Co.,  of  Hartford,  Ct. 


SECTION   II. 

OF    TIMBER    MEASURE. 


1,  What  methods  have  already  been  explained? 

The  methods  of  finding  both  the  superficial  content  of 
boards  and  the  solid  content  of  timber,  by  rules  and  scales, 
have  already  been  given.  We  shall  now  give  the  more 
accurate  methods  by  means  of  figures. 

2.  How  do  you  Jind  the  area  of  a  board,  or  plank  ? 
Multiply  the  length  by  the  breadth,  and  the  product  will 

be  the  content  required. 

9 


194  BOOK    VI. SECTION    II. 

3.  How  do  you  find  it  when  the  hoard  tapers  ? 

If  the  board  is  tapering,  add  the  breadths  of  the  two  ends 
together,  and  take  half  the  sum  for  a  mean  breadth,  and 
multiply  the  result  by  the  length. 

4.  How  may  the  examples  he  done  ? 

The  examples  may  either  be  done  by  cross  multiplica- 
tion, or  the  inches  may  be  reduced  to  the  decimals  of  a 
foot,  and  the  numbers  then  multiplied  together. 

EXAMPLES. 

1.  What  is  the  area  of  a  board  whose  length  is  8  feet' 
6  inches,  and  breadth  1  foot  3  inches  1 


j?y  cross  multiplication, 
ft.  in. 
8    6 

1  3 
8    6'' 

2  \    6'' 


10    7'  6''  content. 


By  decimals, 
ft.  in. 
8    6   =8.  5  ft.      ' 
1    3"=zl.25^: 
Product  =  ia.625  sq.ft. 


2.  What  is  the  content  of  a  board  12  feet  6  inches  long, 
and  2  feet  3  inches  broad? 

Ans.  28  ft.  y  6'',  or  28.125  sq.ft. 

3.  How  many  square  feet  in  a  board  whose  breadth  at 
one  end  is  15  inches,  at  the  other  17  inches,  the  length 
of  the  board  being  6  feet?  .        g 

4.  How  many  square   feet  in   a   plank  whose   length  is 

20  feet,  and  mean  breadth  3  feet  3  inches  ? 

•    .  '  Ans.  65. 

5.  What  is  the  value  of  a  plank  whose  breadth  at  one 
end  is  2  feet,  and  at  the  other  4  feet,  the  length  of  the 
plank  being  12  feet,  and  the  value  per  square  foot  10  cents  ? 

A.ns.  $3.60. 


OF    TIMBER    MEASURE.  195 

•  5.  Having  given  one  dimension  of  a  plank  or  board,  how 
do  you  find  the  other  dimension  such,  that  the  plank  shall 
contain  a  given  area  ? 

Divide  the  given  area  by  the  given  dimension,  and  the 
quotient  will  be  the  other  dimension. 

EXAMPLES. 

1.  The  length  of  a  board  is  16  feet;  what  must  be  its 
width  that  it  may  contain  13  square  feet? 

16  feet  —  192  inches 

12  square  feet  =;  144  x  12  =:  1728  square  inches. 
Then,   1728  ^  192  =  9   inches,  the  width  of  the  4)oard. 

2.  If  a  board  is  6  inches  broad,  what  length  must  be 
cut  from  it  to  make  a  square  foot?  .        „   /> 

3.  If  a  board  is  8  inches  wide,  what  length  of  it  will 
make  4  square  feet?  ^^^    g   r^ 

4.  A  board  is  5  feet  3  inches  long;  what  width  will 
make  7  square  feet?  ^^^    ^  ^^  4  -^ 

5.  What  is  the  content  of  a  board  whose  length  is  5 
feet  7  inches,  and  breadth  1  foot  10  inches? 

Ans.   10  ft.  2'  W. 

6.  How  do  you  fnd  the  solid  content  of  squared  or  four- 
sided  timber  which  does  not  taper? 

Multiply  the  breadth  by  the  depth,  and  then  multiply  the 
product  by  the  length :  the  result  will  be  the  solid  content. 

EXAMPLES. 

1.  A  squared  piece  of  timber  is  15  inches  broad,  15 
inches  deep,  and  18  feet  long:  how  many  solid  feet  does 
it  contam?  iln*.  28.125. 


196  BOOK    VI. SECTION    II. 

2.  What  is  the  solid  content  of  a  piece  of  timber  whose 
breadth  is  16  inches,  depth  12  inches,  and  length  12  feet? 

Ans.  16  ft. 

3.  The  length  of  a  piece  of  timber  is  24.5  feet ;  its  ends 

are   equal  squares,  whose  sides   are  each  1.04  feet:   what 

is  the  solidity  ? 

Ans.  26.4992  solid  ft. 

7.  How  do  you  find  the  solidity  of  a  squared  piece  of  tim- 
ber which  tapers  regularly  ? 

1st.  Add  together  the  breadths  at  the  two  ends,  and  also 
the  depths. 

2d.  Multiply  these  sums  together,  and  to  the  result  add 
the  products  of  the  depth  and  breadth  at  each  end. 

3d.  Multiply  the  last  result  by  the  length,  and  take  one- 
sixth  of  the  product,  which  will  be  the  solidity. 

EXAMPLES. 

1.  How  many  cubic  feet  in  a  piece  of  timber  whose 
ends  are  rectangles,  the  length  and  breadth  of  the  larger 
being  14  inches  and  12  inches;  and  of  the  smaller,  6  and 
4  inches,  the  length  pf  the  piece  being  30i  feet  ? 

.    14  12  16  X  20  =  320 

_6  _4  14  X  12  =:  168 

20  16  6x4  =_24 

612  square  inches. 

But,  512  square  inches  =  ^-a   square  feet. 
Then,     sj  x  30i  X  |  =  IS/-^  solid  feet. 

2.  How  many  solid  inches  in  a  mahogany  log,  the  depth 
and  breadth  at  one  end  being  81  ^  inches  and  55  inches, 
and  of  th«  other  41  and  29^  inches,  the  length  of  the  log 
being  47^  laches? 

Ans.   126340.59375. 


OF    TIMBER    MEASURE.  197 

3.  How  many  cubic  feet  in  a  stick  of  timber  whose 
larger  end  is  25  feet  by  20,  the  smaller  15  feet  by  10, 
and  the  length  12  feet? 

Ans.  3700. 

4.  What  is  the  number  of  cubic  feet  in  a  stick  of  hewn 
timber,  whose  ends  are  30  inches  by  27  and  24  inches 
by  18,  the  length  being  24  feet? 

Ans.  102. 

5.  The  length  of  a  piece  of  timber  is  20.38  feet,  and  the 
ends  are  unequal  squares:  the  side  of  the  greater  is  19| 
inches,  and  of  the  less  9^  inches :  what  is  the  solid  content  ? 

Ans.  30.763  cubic  ft. 

6.  The  length  of  a  piece  of  timber  is  27.36  feet:  at 
the  greater  end,  the  breadth  is  1.78  feet  and  the  thickness 
1.23  feet;  and  at  the  less  end,  the  breadth  is  1.04  feet 
and  the  thickness  0.91  feet:   what  is  its  solidity? 

Ans.  41.8179  cubic  ft. 

8.  How  do  you  do  when  the  timber  does  not  taper  regu- 
larly ? 

If  the  timber  does  not  taper  regularly,  measure  parts  of 
the  stick,  the  same  as  if  it  had  a  regular  taper,  and  take 
the  sum  of  the  parts  for  the  entire  solidity. 


9.  Knowing  the  area  of  the  end  of  a  square  piece  of  tim- 
ber which  does  not  taper,  how  do  you  find  the  length  which 
must  be  cut  off  in  order  to  obtain  a  given  solidity?  . 

1st.  Reduce  the  given  solidity  to  cubic  inches. 

2d.  Divide  the  number  of  cubic  inches  by  the  area  of 
the  end  expressed  in  inches,  and  the  quotient  will  be  the 
length  in  inches. 


198  BOOK    VI. SECTION    II. 

EXAMPLES. 

1.  A  piece  of  timber  is  10  inches  square  :  how  much 
must  be  cut  off  to  make  a  sqlid  foot  ? 

10  X  10  =  100  square  inches. 
Then,  1728  -^  100  =  17.28  inches. 

2.  A  piece  of  timber  is  20  inches  broad  and  10  inches 
deep:   how  much  in  length  will  make  a  solid  foot? 

Ans.  8^f  in. 

3.  A  ^iece  of  timber  is  9  inches  broad  and  6  inches 
deep :   how  much  in  length  will  make  3  solid  feet  ? 

Ans.  S  ft. 

10.  How  do  you  find  the  solidity  of  round  or  unsquared 
timber  1 

1st.  Take  the  girt  or  circumference,  and  then  divide  it 
by  5. 

2d.  Multiply  the  square  of  one-fifth  of  the  girt  by  twice 
the  length,  and  the  product  will  be  the  solidity  very  nearly. 

EXAMPLES. 

1.  A  piece  of  round  timber  is  9|  feet  in  length,  and  the 
girt  is  13  feet:  what  is  its  solidity? 

First,  13  H-  5  =r  2.6  the  fifth  of  the  girt. 
Also,  2^'  =  6.76  ;  and  9.75-  x  2  =z  19.50. 
Again,       6.76  x  19.5  —  131.82  cubic  feet,  which  is  the 
required  solidity. 

2.  The  length  of  a  tree  is  24  feet,  and  the  girt  through- 
out 8  feet :    what  is  the  content  ? 

Ans.  122.88  cuhic  ft.   . 

3.  Required  the  content  of  a  piece  of  timber,  its  length 
being  9  feet  6  inches,  and  girt  14  feet. 

Ans.  148.96  cmWc/^ 


OF    LOGS    FOR    SAWING.  199 

11,  Hew  do  you  do  when  tlw  timber  tapers? 

Gird  the  timber  at  as  many  points  as  may  be  neces- 
sary, and  divide  the  sum  of  the  girts  by  their  number  for 
the  mean  girt,  of  which  take  one-iifth,  and  proceed  as 
before. 

4.  If  a  tree,  girt   14  feet  at  the  thicker  end  and  2  feet 

at  the  smaller  end,  be  24  feet  in  length,  how  many  solid 

feet  will  it  contain? 

Ans.  122.88. 

5.  A  tred   girts  at   five  different  places   as   follows :    in 

the  first  9.43  fe^t;   in  the  second  7.92  feet;   in  the  third 

6.15  feet;    in  the   fourth  4.74  feet;    and  in  the  fifth  3.16 

feet:  now,  if  the  length  of  the  tree  be  17.25  feet,  what  is 

its  solidity? 

Ans.  54:A2499  cubic  ft. 


OF    LOGS    FOR    SAWING. 

12.  What  is  often  necessary  for  lumber  merchants? 

It  is  often  necessary  for  lumber  merchants  to  ascertain 
the  number  of  feet  of  boards  which  can  be  cut  from  a 
given  log ;  or,  in  other  words,  to  find  how  many  logs  will 
be  necessary  to  make  a  given  amount  of  boards. 

13.  What  is  a  standard  board? 

A  standard  board  is  one  which  is  12  inches  wide,^  1 
inch  thick,  arid  12  feet  long :  hei^ce,  a  standard  board  is 
1  inch  thick  and  contains  12  square  feet. 

14.  What  is  a  standard  saw-log  ? 

A  standard  log  is  12  feet  long  and  24  inches  in  diam- 
eter. 


200  BOOK    VT. SECTIt)N    II. 

15.  How  wiil  you  find  the  number  of  feet  of  boards  which, 
can  be  sawed  from  a  standard  log  ? 

If  we   saw  off,  say  2   inches,  from   each  side,  the   log 

will  be  reduced  to  a  square   20  inches  on  a  side.     Now, 

since  a  standard  board  is  one  inch  in  thickness,  and  since 

the  saw  cuts   about  one   quarter  of  an  inch  each  time  it 

goes  through,  it  follows  that  one -fourth  of  the  log  will  be 

consumed  by  the  saw.     Hence  we  shall  have 

3 
20  X  —  =  the  number  of  boards  cut  from  the  log. 

Now,  if  the  width  of  a  board  in  inches  be  divided  by  12, 

and  the  quotient  be   multiplied  by  the   length  in  feet,  the 

product  will  be  the   number  of  square   feet  in   the    board. 

.  20 
Hence,    —  X  length  of  the  log  in  feet  =  the  square  feet 

in  each  board.     Therefore, 

3        20 
20  X  —  X  —  X  length  of  log  =  the   square  feet  in  all 

the  boards, 

3*2  1 

=  20xl0x—  X  —  X  lengthof  log  =  20  x  10  X  —  X 
4  1^  o 

length ;    and  the   same   may  be   shown   for  a    log  of  any 

length. 

16.  What  then  is  the  rule  for  finding  the  number  offset 
of  hoards  which  can  be  cut  from  any  log  whatever  ? 

From  the  diameter  of  the  log,  in  inches,  subtract  4  for 
the  slabs.  Then  multiply  the  remainder  by  half  itself  and 
the  product  by  the  length  of  the  log,  in  feet,  and  divide  the 
result  by  8  :  the  quotient  will  be  the  number  of  square  feet. 

EXAMPLES. 

1.  What  is  the  number  of  feet  of  boards  which  can  be 
cut  from  a  standard  log? 


bricklayers'  work.  201 


Diameter 

24 

inches 

for  slabs 

4 

remainder 

-20-- 

half  remainder        10 

200 

length  of  log 

12 
8)2400 

300  =  the  number  of  feet. 

2.  How  many  feet  can  be  cut  from  a  log  12  inches  in 
diameter  and  12  feet  long?  .  a^    ^g 

*    3.  How  many  feet  can  be  cut  from  a  log  20  inches  in 
diameter  and  16  feet  long?  ,        25R 

4.  How  many  feet  can  be  cut  from  a  log  24  inches  in 
diameter  and  16  feet  long?  .         .qq 

5.  How  many  feet  can  be  cut  from  a  log  28  inches  in 
diameter  and  14  feet  long?  .        ^q^ 


SECTION   III. 


BRICKLAYERS     WORK. 


1,  In  how  many  ways  is  artificers^  work  computed? 

Artificers'  work,  in  general,  is  computed  by  three  differ- 
ent measures,  viz. : 

1st.  The  linear  measure,  or,  as  it  is  called  by  mechanics, 
running  measure. 

2d.  Superficial  or  square  measure,  in  which  the  compu- 
tation is  made  by  the  square  foot,  square  yard,  or  by  the 
square  containing  100  square  feet,  or  yards. 

9* 


202  BOOK    VI. SECTION    III. 

3d.  By  the  cubic  or  solid  measure,  when  it  is  estimated 
by  the  cubic  foot,  or  the  cubic  yard.  The  work,  however, 
is  often  estimated  in  square  measure,  and  the  materials  for 
construction  in  cubic  measure. 

2.  What  proportion  do  the  dimensions  of  a  brick  bear  to 
each  other? 

The  dimensions  of  a  brick  generally  bear  the  following 

proportions  to  each  other,  viz. : 

Length  =  twice  the  width,  and 
Width   =  twice  the  thickness,  and 

hence,  the  length  is  equal  to  four  times  the  thickness.     . 

3.  What  are  the  common  dimensions  of  a  brick  ?  How 
many  cubic  inches  does  it  contain  ? 

The  common  length  of  a  brick  is  8  inches,  in  which 
case  the  width  is  4  inches,  and  the  thickness  2  inches. 
A  brick  of  this  size  contains 

8  X  4  X  2  =  64  cubic  inches ;  and  since  a  cubic  foot 
contains  1728  cubic  inches,  we  have 

1728  -f-  64  =  27  the  number  of  bricks  in  a  cubic  foot. 

^.  If  a  brick  is  9  inches  long,  what  will  be  its  width  and 
what  its  content  ? 

If  the  brick  is  9  inches  long,  then  the  width  is  4^  inches, 
and  the  thickness  2^ ;  and  then  each  brick  will  contain 

9  X  4^  X  21  =:  91|  cubic  inches  in  each  brick;  and 
1728  ^911  =  19  nearly,  the  number  of  bricks  in  a  cubic 

foot.     In  the  examples  which  follow,  we  shall  suppose  the 
brick  to  be  8  inches  long. 

5,  How  do  you  find  the  number  of  bricks  required  to  build 
a  wall  of  given  dimensions  ? 

1st.  Find  the  content  of  the  wall  in  cubic  feet. 


bricklayers'  work.  203 

2d.  Multiply  the  number  of  cubic  feet  by  the  number  of 
bricks  in  a  cubic  foot,  and  the  result  will  be  the  number 
of  bricks  required. 

EXAMPLES. 

1.  How  many  bricks,  of  8  inches  in  length,  will  be  re- 
quired to  build  a   wall  30   feet  long,  a  brick  and  a  half 

thick,  and  15  feet  in  height? 

Ans.  12150. 

2.  How  many  bricks,  of  the  usual  size,  will  be  required 
to  build  a  wall  50  feet  long,  2  bricks  thick,  and  36  feet 

6.  What  allowance  is  made  for  the  thickness  of  the  mortar  ? 
The  thickness  of  mortar  between  the  courses  is  nearly 

a  quarter  of  an  inch,  so  that  four  courses  will  give  nearly 
1  inch  in  height.  The  mortar,  therefore,  adds  nearly 
one-eighth  to  the  height;  but  as  one-eighth  is  rather  too 
large  an  allowance,  we  need  not  consider  the  mortar  which 
goes  to  increase  the  length  of  the  wall. 

3.  How  many  bricks  would  be  required  in  the  first  and 

second  examples,  if  we  make    the   proper  allowance   for 

mortar  ? 

V      ^^^    ^  1st,  106311. 

,      ^^'   i  2d.  56700. 

7.  How  do  bricklayers  generally  estimate  their  work? 
Bricklayers   generally  estimate  their  work  St  so    much 

per  thousand  bricks.  To  find  the  value  of  things  estimated 
hy  the  thousand,  see  Arithmetic,  page  192, 

4.  What  is  thp  cost  of  a  wall  60  feet  long,  20  feet  high, 
and  two  and  a  half  bricks  thick,  at  $7.50  per  thousand, 
which  price  we  suppose  to  include  the  cost  of  the  mortar? 


204  BOOK    VI.  — SECTION    III. 

If  we  suppose  the  mortar  to  occupy  a  space  equal  to 
one-eighth  the  height  of  the  wall,  we  must  find  the  quantity 
of  bricks  under  the  supposition  that  the  wall  was  17i  feet 

^^  ^^^g^*-  '  .  Ans.  8354.371. 

8,  In  estimating  the  bricks  for  a  house,  what  allowances 
are  made? 

In  estimating  the  bricks  for  a  house,  allowance  must  be 
made  for  the  windows  and  doors. 


OF    CISTERNS. 

9.  What  are  cisterns  ?  -  - 
Cisterns  are    large  reservoirs  constructed  to  hold  water, 

and  to  be   permanent,   should  be   made   either  of  brick  or 
masonry. 

It  frequently  occurs  that  they  are  to  be  so  constructed 
as  to  hold  given  quantities  of  water,  and  it  then  becomes 
a  useful  and  practical  problem  to  calculate  their  exact 
dimensions.        , 

10.  How  many  cubic  inches  in  a  hogshead? 

It  was  remarked  in  Arithmetic,  page  104,  that  a  hogs- 
head contains  63  gallons,  and  that  a  gallon  contains  231 
cubic  inches.  Hence,  231  X  63  =  14553,  the  number  of 
cubic  inches  in  a  hogshead. 

11.  How  do  you  find  the  number  of  hogsheads  which  a 
cistern  of  given  dimensions  will  contain  ? 

}«t.  Find  the  solid  content  of  the  cistern  in  cubic  inches. 
2d.  Divide  the  content  so  found  by  14553,  and  the  quo- 
tient will  be  the  number  of  hogsheads. 


OF    CISTERNS. 


example;. 


OF  til.. 


The  diameter  of  a  cistprii  is  6  fefet  6  inches,  and  height 
10  feet:  how  many  hogsheads  does  it  contain? 

The  dimensions  reduced  to  inches  are  78  and  120.  To 
find  the  solid  content,  see  page  162.  Then,  the  content 
in  cubic  inches,  which  is  573404.832,  gives 

573404.832  -f-  14553  =.39.40  hogsheads,  nearly. 

12,  If  the  height  of  a  cistern  be  given,  how  do  you  find 
the  diameter,  so  that  the  cistern  shall  contain  a  given  number 
of  hogsheads  ? 

1st.  Reduce  the  height  of  the  cistern  to  inches,  and  the 
content  to  cubic  inches. 

2d.  Multiply  the  height  by  the  decimal  .7854. 

3d.  Divide  the  content  by  the  last  result,  and  extract 
the  square  root  of  the  quotient,  which  will  be  the  diameter 
of  the  cistern  in  inches. 

EXAMPLE. 

The  height  of  a  cistern  is  10  feet:  what  must  be  its 
diameter,  that  it  may  contain  40  hogsheads  ? 

Ans.  78.6  in.  nearly. 

13.  If  the  diameter  of  a  cistern  be  given,  how  do  you  find 
the  height,  so  that  the  cistern  shall  contain  a  given  number 
of  hogsheads  ? 

1st.  Reduce  the  content  to  cubic  inches. 

2d.  Reduce  the  diameter  to  inches,  and  then  multiply  its 
square  by  the  decimal  .7854. 

3d.  Divide  the  content  by  the  last  result,  and  the  quo- 
tient will  be  the  height  in  inches. 


206  BOOK    VI. SECTION    IV. 

EXAMPLE. 

The  diameter  of  a   cistern  is  8  feet :  what  must  be  its 
height  that  it  may  contain  150  hogsheads  ? 

Ans.  25  ft.  1  in.  nearly. 


SECTION   IV. 

masons'  work. 


1.  What  belongs  to  masonry,  and  what  measures  are  used? 

AH  soi;ts  of  stone  work.  The  measure  made  use  of  is 
either  superficial  or  soHd. 

Walls,  columns,  blocks  of  stone  or  marble,  are  measured 
by  the  cubic  foot;  and  pavements,  slabs,  chimney-pieces, 
&c.,  are  measured  by  the  square  or  superficial  foot.  Cubic 
or  solid  measure  is  always  used  for  the  materials,  and  the 
square  measure  is  sometimes  used  for  the  workmanship. 

EXAMPLES. 

1.  Required  the  solid  content  of  a  wall  53  feet  6  mches 
long,  12  feet  3  inches  high,  and  2  feet  thick. 

Ans.  1310f /^. 

2.  What  is  the  solid  content  of  a  wall,  the  length  of 
which  is  24  feet  3  inches,  height  10  feet  9  inches^  and 
thickness  2  feet?  ^^^    521.375 /^ 

3.  In  a  chirnney-piece  we  find  the  following  dimensions: 
Length  of  the  mant«l  and  slab,     4  feet    2  inches. 
Breadth  of  both  together,  3     "      2       " 
Length  of  each  jamb,  4     «      4       « 
Breadth  of  both,  1     "       9       " 
Required  the  superficial  content. 

Ans.  21  ft.  l(y. 


carpenters'  and  joiners'  work.  207 

SECTION  V. 

carpenters'  and  joiners'  work. 

1,  In  what  does  carpenters'  and  joiners''  work  consist  1 
Carpenters'  and  joiners'  work  is  that  of  flooring,  roofing, 

&c.,  and  is  generally  measured  by  the  square  of  lOQ  square 

feet. 


2. .  When  is  a  roof  said  to  have  a  true 

In  carpentry,  a  roof  is  said  to  have  a  true  -pitch  when 
the  length  of  the  rafters  is  three-fourths  the  breadth  of  the 
building.  The  rafters  then  are  nearly  at  right  angles.  It 
is  therefore  customary  to  take  once  and  a  half  times  the 
area  of  the  flat  of  the  building  for  the  area  of  the  roof. 

EXAMPLES. 

1.  How  many  squares,  of  100  square  feet  each,  in  a 
floor  48  feet  6  inches  long,  and  24  feet  3  inches  broad? 

Ans.   1 1  and  76^  sq.  ft, 

2.  A  floor  is  36  feet  3  inches  long,  and  16  feet  6  inches 
broad :  how  many  squares  does  it  contain  ? 

Ans.  5  and  98|  sq.  ft.  . 

3.  How  many  squares  are  there  in  a  partition  91  feet 
9  inches  long,  and  11  feet  3  inches  high? 

Ans.   10  and  32  sq.ft. 

4.  If  a  house  measure  within  the  walls  52  feet  8  inches 
in  length,  and  30  feet  6  inches  in  breadth,  and  the  roof  be 
of  the  true  pitch,  what  will  the  roofing  cost  at  $1.40  per 

«1»^'«-  Ans.  «33.733. 


208  BOOK    VI. SECTION    V. 

OF    |JNS    FOR    GRAIN. 

,3.    What  is  a  bin  1 

It  is  a  wooden  box  used  by  farmers  for  the  storage  of 
their  grain. 

4.  Of  what  form  are  bins  generally  made  ? 

Their  bottoms  or  bases  are  generally  rectangles,  and 
horizontal,  and  their  sides  vertical. 

5.  How  many  cubic  feet  are  there  in  a  bushel? 

Since  a  bushel  contains  2150.4  cubic  inches,  (see  Arith- 
metic, page  106,)  and  a  cubic  foot  1728  inches,  it  follows 
that  a  bushel  contains  one  and  a  quarter  cubic  feet,  nearly. 

6.  Having  any  number  of  bushels,  how  then  will  you  find 
the  corresponding  number  of  cubic  feet  ? 

Increase  the  number  of  bushels  bne-fourth  itself,  and  the 
result  will  be  the  number  of  cubic  feet. 

EXAMPLES. 

1 .  A  bin  contains  372  bushels ;  how  many  cubic  feet 
does  it  contain? 

372  H-  4  rn  93  ;  hence,  372  +  93  =  465  cubic  feet. 

2.  In  a  bin  containing  400  bushels,  how  many  cubic 
feet?  Ans.  500. 

7.  How  tbill  you  find  the  number  of  bushels  -which  a  bin 
of  a  given  size  will  hold  ? 

Find  the  content  of  the  bin  in  cubic  feet ;  then  diminish 
the  content  by  one-fifth,  and  the  result  will  be  the  content 
in  bushels. 

3.'  A  bin  is  8  feet  long,  4  feet  wide,  and  5  feet  high  ^ 
how  many  bushels  will  it  hold? 


»    "of  bins  for  grain.  20D 

8  X  4  X  5  =  160 
then,     160  H-  5  =    32  :   160  —  32  =  128  bushels  = 
capacity  of  bin. 

4.  How  many  bushels  will  a  bin  contain  which  is  7  feet 
long,  3  feet  wide,  and  6  feet  in  height  ? 

Ans.   100.8  bush. 

8.  How  will  you  find  the  dimensions  of  a  bin  which  shall 
contain  a  given  number  of  bushels  ? 

Increase  the  number  of  bushels  one-fourth  itself,  and  the 
result  will  show  the  number  of  cubic  feet  which  the  bin 
will  contain.  Then,  when  two  dimensions  of  the  bin  are 
known,  divide  the  last  result  by  their  product,  and  the  quo- 
tient will  be  the  other  dimension. 

5.  What  inust  be  the  height  of  a  bin  that  will  contain 
600  bushels,  its^  length  being  8  feet  and  breadth  4  ? ' 
600  -f-  4  =  150  ;  hence,  600  +  150  =  750  ==  the  cubic  feet ; 
and    8x4=  32,    the    product   of  the    given   dimensions. 
Then,  750  -^  32  =  23.44  feet,  the  height  of  the  bin. 

6.  What  must  be  the  width  of  a  bin  that  shall  contain 
900  bushels,  the  height  being  12  and  the  length  10  feet? 
900  ^4  =  225  ;  hence,  900  +  225  =  1125  rz:  the  cubic  feet ; 
and  12  x  10  =  120,  the  product  of  the  given  dimensions. 
Then,  1125  ^  120  z=z  9.375  feet,  the  width  of  the  bin. 

7.  The  length  of  a  bin  is  4  feet,  its  breadth  5  feet  6 

inches:   what  must  be  its  height  that  it  may  contain  136 

bushels  1  ^        «   /!.   o  •      . 

Ans.  7  ft.  8  in.  + 

8.  The  depth  of  a  bin  is  6  feet  2  inches,  the  breadth 
4  feet  8  inches :  what  must  be  the  length  that  it  may  con- 
tain 200  bushels?  .        ,^,  .      , 

Ans,  104  m.  -f- 


210  BOOK   VI. — SECTION   VII. 

SECTION   VI. 

slaters'  and  tilers'  ■work. 

1.  How  is  the  content  of  a  roof  found? 
,In  this  work,  the  content  of  the  roof  is  found  by  multi- 
plying the   length  of  the  ridge  by  the  girt  from  eaves  to 
eaves.     Allowances,  however,  must  be  made  for  the  double 
rows  of  slate  at  the  bottom. 

EXAMPLES. 

1.  The  length  of  a  slated  roof  is  45  feet  9  inches,  and 
its  girt  34  feet  3  inches:  what  is  its  content? 

•       '         Ans.   1566.9375  sq.ft. 

2.  What  will  tne    tiling  of  a  barn   cost,   at  $3.40  per 

square  of  100  feet,  the   length  being  43    feet    10    inches, 

and  breadth  27  feet  5  inches,  on  the  flat,  the  eave-board 

projecting  16  inches  on  each  side,  and  the  roof  being  of 

the  true  pitch? 

Ans,  865.26. 


SECTION   VII. 

plasterers'  work. 


1,  How  many  kinds  of  plasterers^  work  are  there,  and  how 
are  they  measured? 

Plasterers'  work  is  of  two  kinds,  viz. :  ceiling,  which  is 
plastering  On  laths ;  and  rendering,  which  is  plastering  on 
walls.     These  are  measured  separately. 


plasterers'  wortc.  211 

The  contents  are  estimated  either  by  the  square  foot, 
the  square  yard,  or  by  the  square  of  100  feet. 

Inriched  mouldings,  &c.,  are  rated  by  the  running  or 
liiieal  measure. 

In  estimating  plastering,  deductions  are  made  for  chim- 
neys, doors,  windows,  &c. 

EXAMPLE^. 

1.  How  many  square  yards  are  contained  in  a  ceiling 
43  feet  3  inches  long,  and  25  feet  6  inches  broad  ? 

Ans.   1221  nearly. 

2.  What  is  the  cost  of  ceiling  a  room  21  feet  8  inches, 
by  14  feet  10  inches,  at  18  cents  per  square  yard? 

Ans.  $6,421. 
,3.  The  Ifength  of  a  room^  is  14  feet  5  inches,  breadth 
13  feet  2  inches,  and  height  to  the  under  side  of  the  cor- 
nice 9  feet  3  inches.  The  cornice  girts  8i  inches,  and 
projects  5  inches  from  the  wall  on  the  upper  part  next  the 
ceiling,  deducting  only  for  one  door  7  feet  by  4 :  what  will 
be  the  amount  of  the  plastering? 

r  53  yds.  5  ft.  3^  6''^  of  rendering. 
Ans.    <  l&  yds.  5  ft.  6^  4^^  of  ceiling. 
'(37  ft.  10^  9''  of  cornice. 

How  is  the  area  of  the  cornice  found  in  the  '  above  exam- 
pies? 

^he  mean  length  of  the  cornice,  both  in  the  length  and 
breadth  of  the  house,  is  found  by  taking  the  middle  line  of 
the  cornice.  Now,  since  the  cornice  projects  5  inches  at 
the  ceiling,  it  will  project  2i  inches  at  the  middle  line  ; 
and  therefore,  the  length  of  the  middle  line  along  the  length 
of  the  room  will  be  14  feet,  and  across  the  room,  12  feet 
9  inches.  Then  multiply  the  double  of  each  of  these  num- 
bers by  the  girth,  which  is  81  inches,  and  the  sum  of  the 
products  will  be  the  area  of  the  .cornice. 


212  BOOK    VI. SECTION    IX. 

SECTION  VIII. 
painters'  work. 

How  is  painters'  work  computed,  and  what  allowances  are 
made  ? 

Painters'  work  is  computed  in  square  yards.  Every  part 
is  measured  where  the  color  lies,  and  the  measuring  line 
is  carried  into  all  the  mouldings  and  cornices. 

Windows  are  generally  done  at  so  much  a  piece.  It  is 
usual  to  allow  double  measure,  for  tarved  mouldings,  &c. 

EXAMPLES. 

1.  How  hlany  yards  of  painting  in  a  room  which  is  65 
feet  6  inches  in  perimeter,  and  12  feet  4  inches  in  height  ? 

Ans.  89f }  sq.  yds. 

2.  The  length  of  a  room  is  20  feet,  its  breadth  14  feet 
6  inches,  and  height  10  feet  4  inches:  how  many  yards 
of  painting  are  in  it,  deducting  a  fire-place  of  4  feet  by 
4  feet  4  inches,  and  two  windows,  each  6  feet  by  3  feet 
2  inches?  Ans.  73^^  sq.  yds. 


SECTION   IX. 
pavers'  work. 


ffow  is  pavers^  work  estimated? 

Pavers'  work  is  done  by  the  square  yard,  and  the  con- 
tent is  found  by  multiplying  the  length  and  breadth  to- 
gether. 

EXAMPLES. 

1.  What  is  the   cost  of  paving  a  side-walk,  the  length 


PLUMBERS     WORK. 


213 


of  which  is  35  feet  4  inches,  and  breadth  8  feet  3  inches, 
at  54  cents  per  square  yard?  ^^^    ^^^^g  ^ 

2.  What  will  be  the  cost  of  paving  a  rectangular  court- 
yard, whose  length  is  63  feet,  and  breadth  45  feet,  at  2^. 
66?.  per  square  yard ;  there  being,  however,  a  walk  running 
lengthwise  5  feet  3  inches  broad,  which  is  to  be  flagged 
with  stone  costing  3  shillings  per  square  yard? 

Ans.  £40  5s.  lO^d. 


SECTION   X. 


EJ.UBJBERS'    WORK. 

1.  Plumbers'  work  is  rat^d  at  so  much  a  pound,  or  else 
by  the  hundred  weight.  Sheet  lead,  used  for  gutters,  &c., 
weighs  from  6  to  12  lbs.  per  square  foot.  Leaden  pipes 
vary  in  weight  according  to  the  diameter  of  their  bore  and 
thickness. 

The  following  table  shows  the  weight  of  a  square  foot 
of  sheet  lead,  according  to  its  thickness ;  and  the  common 
weight  of  a  yard  of  leaden  pipe,  according  to  the  diameter 
of  the  bore. 


Thickness 
of  lead. 

Pounds  to  a  square 
foot. 

Bore  of 
leaden  pipes. 

Pounds 
per  yard. 

Inch. 

tV 

5.899 

Inch. 

Of 

10 

i 

6.554 

1 

12 

i 

7.373 

n 

16 

+ 

8.427 

H 

18 

i 

9.831 

If 

21 

i 

11.797 

2 

24 

214  BOOK    VI. SECTION   X. 

^  EXAMPLES.. 

1.  What  weight  of  lead  of  ^^  of  an  inch  in  thickness, 
will  cover  a  flat  15  feet  6  inches  long,  and  10  feet  3  inches 
broad,  estimating  the  weight  at  6  lbs.  per  square  foot? 

^^    i  .      •  -^ns.  8  cwt.  2  qr.  \\lh. 

2.  What  will  be  the  cost  of  130  yards  of  leaden  pipe 
of  an  inch  and  a  half  bore,  at  8  cents  per  pound,  sup- 
posiAg  each  yard  to  weigh  18  lbs.? 

Ans.   $187.20. 

3.  The  lead  used  for  a  gutter  is  12  feet  5  inches  long 
and  1  foot  3  inches  broad :  what  is  its  weight,  supposing  it 
to  be  \  of  an  inch  in  thickness  ? 

Ans.   101  lbs.  12  oz.  13.6  dr. 

4.  What  is  the  weight  of  96  yards  of  leaden  pipe,  of 
an  inch  and  a  quarter  bore  ? 

Ans.   13  cwt.  2  qr.  24  lbs. 

,5.  What  will  be  the  cost  of  a,  sheet  of  lead  16  feet  6 
inches  long  and  10  feet  4  inches  broad,  at  5  cents  per 
pound;  the  lead  being  \  of  an  inch  in  thickness  ? 

Ans.  883.81. 


OF    MATTER    AND    BODIES.  215 


BOOK  VII. 


INTRODUCTION    TO    MECHANICS. 

SECTION   I. 

OF    MATTER    AND    BODIES. 

1.  What  is  matter? 

Matter  is  a  general  name  for  every  thing  which  has 
substance,  and  is  always  capable  of  being  increased  or  di- 
minished. Whatever  we  can  touch,  taste,  smell,  or  see,  is 
matter. 

2.  What  is  a  body? 

A  BODY  is  any  portion  of  matter. 

3.  What  is  space  ?     How  many  dimensions  has  it  ? 

Space  is  mere  extension,  in  which  all  bodies  are  sit- 
uated. Thus,  when  a  body  has  a  certain  place,  it  is  said 
to  occupy  that  portion  of  space  which  it  fills.  Space  has 
three  dimensions— length,  breadth,  and  thickness.      '  "-^ 

4.  What  are  the  properties  common  to  all  bodies  ? 

The  properties  which  belong  to  all  bodies  are:  impenr- 
trability,  extension^  figure,  divisibility,  inertia,  and  attraction. 

5.  What  is  impenetrability? 

Impenetrability  is  the  property,  in  virtue  of  which  a 


216  BOOK    VII. SECTION    I. 

body  must  fill  a  certain  space,  and  which  no  other  body 
can  occupy  at  the  same  time.  Thus,  if  you  fill  a  vessel 
full  of  water,  and  then  plunge  in  your  hand,  or  a  stick, 
some  of  the  water  will  be  forced  over  the  top  of  the  ves- 
sel. Your  hand  or  the  stick  removes  the  water,  and  does 
not  occupy  the  space  until  after  the  water  is  displaced. 

6.  What  is  extension  ?        - 

Since  a  body  occupies  space,  it  must,  like  any  portion 
of  space,  have  the  dimensions  of  length,  breadth,  and  thick- 
ness. These  are  called  the  dimensions  of  extension,  ^nd 
vary  in  different  bodies.  The  length,  breadth,  and  depth 
of  a  house  are  very  different  from  those  of  an  inkstand. 

7.  How  are  length  and  breadth  measured  1  How  do  you 
mffasure  height  and  depth  ? 

Length  and  breadth  are  generally  measured  in  a  hori- 
zontal direction.  Height  and  depth  are  the  same  dimen- 
sion :  height  is  measured  upward,  and  depth  downward. 
Thus,  we  say  a  mountain  is  400  feet  high,  and  a  river  50 
feet  deep. 

S.    ^Vhat  is  figure  ? 

Figure  is  merely  the  limit  of  extension.  Figiure  is  also 
called  forin  or  shape. 

9.  When  is  a  body  said  to  be  regular  ?    when  irregular  1 
If  all  the  parts  of  a  body  are  arranged  in  the  same  way, 

about  a  line  or  a  centre,  the  body  is  said  to  be  regular  or 
symmetrical;  and  when  the  parts  are  not  so  arranged,  the 
body  is  said  to  be  irregular.  Nature  has  given  regular 
forms  to  nearly  all  her  productions. 

10.  What  is  divisibility? 

Divisibility  denotes  the  susceptibility  of  matter  to  be 


OF    MATTER    AND    BODIFS.  217 

continually  divided.  That  is,  a  portion  of  matter  may  be 
divided,  and  each  part  again  divided,  and  each  of  the  parts 
divided  again,  and  so  on,  continually,  without  ever  arriving 
at  a  portion  which  will  be  absolutely  nothing. 

Suppose,  for  instance,  you  take  a  portion  of  matter,  say 
one  pound  or  one  ounce,  and  divide  it  into  two  equal  parts, 
and  then  divide  each  part  again  into  two  equal  parts,  and 
so  on  continually.  Now,  all  the  parts  will  continually  grow 
smaller  and,  smaller, 'but  no  one  of  them  will  ^ev6r  become 
equal  to  nothing,  since  the  half  of  a  thing  must  always 
have  some  value.  • 

IL    What  is  inertia? 

Inertia  is.  the  resistance  which  matter  makes  to  a  change 
of  state.  Bodies'  ar%  not  only  incapable  of  changing  their 
actual  state,  whether  it  be  that  of  motion  or  rest,  but  they 
seem  endowed  with  the  power  of  resisting  such  a  change. 
This  property  is  called  inertia. 

12.  If  a  body  is  at  rest,  will  it  remain  so?  If  in  mo- 
tion, will  it  continue  so  ? 

If  a  body  i§  at  rest  it  will  remain  so,  unless  something 
be  applied  from  without  to  move  it ;  and  if  it  be  moving, 
it  will  contin.ue  to  move,  unless  something  stops'  it. 

13.  What  are  ;atoms  ? 

The  smallest  parts  into  which  we  can  suppose  a  body 
divided,  are'  called  particles  or  atoms. 

14.  Do  these  atoms  adhere  to  each  other  ?   . 
They  do,  and  form  masses  or  bodies. 

15.  What  is  the  force  called  which  unites  them?  ..: 

It  is  called  the  attraction  of  cohesion.     Without  this  power  ' 
solid  bodies  would  crumble  to  pieces  and  fall  to  atoms.^ 

10  ' 


218  BOOK    VII. SECTION    I. 

16.  In  what  kind  of  bodies  does  this  attraction  exist  ? 
In  all  bodies,  fluid  as  well  as  solid.     It  is  the  attraction 

of  cohesion  which  holds  a  drop  of  water  in  suspension  at 
the  end  of  the  finger,  and  causes  it  to  take  a  spherical 
form. 

The  attraction  of  cohesion  is  stronger  in  some  substance? 
than  in  others.  Those  in  which  it  is  the  weakest  are  easily 
broken,  or  the  attraction  is  easily  overcome;  while  those 
in  which  it  is  greater,  are  proportionably  stronger. 

17.  What  is  the  difference  between  the  attraction  of  cohe- 
sion and  the  attraction  of  gravitation  ? 

The  attraction  of  cohesion  unites  the  particles  of  matter, 
and  these  by  their  aggregation  form  masses  or  bodies.  The 
attraction  of  gravitation  is  the  force  by  which  masses  of 
matter  tend  to  come  together.  The  attraction  of  cohesior. 
acts  only  between  particles  of  matter  which  are  very  neav 
each  other,  while  the  attraction  of  gravitation  acts  betweer.. 
bodies  widely  separated. 

18.  Is  the  attraction  between  bodies  mutual? 

The  attraction  between  two  bodies  is  mutual;  that  is, 
each  body  attracts  the  other  just  as  much,  and  no  more, 
than  it  is  attracted  by  it.  But  if  the  bodies  are  left  free, 
the  smaller  will  move  towards  the  larger ;  for,  as  they  are 
urged  together  by  equal  forces,  the  smaller  will  obey  the 
force  faster  than  the  larger.  Thus,  the  earth  being  larger 
than  any  body  near  its  surface,  forces  all  bodies  towards 
it,  and  they  immediately  fall  unless  the  attraction  of  gravi- 
tation is  counteracted. 

It  should,  however,  be  borne  in  mind  that  every  body 
attracts  the  earth  just  as  strongly  as  the  earth  attracts  the 
body ;  and  the  body  moves  towards  the  earth,  only  because 
the  earth  is  larger,  and  therefore  not  as  rapidly  moved  by 
their  mutual  attraction. 


LAWS    OF    MOTION,    ETC.  219 

19.    What  is  weight? 

Weight  is  the  force  which  is  necessary  to  overcome  the 
attraction  of  gravitation.  Thus,  if  we  have  two  bodies, 
and  one  has  twice  as  much  tendency  to  descend  towards 
the  earth  as  the  other,  it  will  require  just  twice  as  much 
force  to  support  it,  and  hence  we  say  that  it  is  twice  as 
heavy. 


SECTION   II. 

LAWS    OF    MOTION,    ANC    CENTRE    OF    GRAVITY. 

1.  What  is  motion? 

Motion  is  a  change  of  place.  Thus,  a  body  is  said  to 
be  in  motion  when  it  is  continually  changing  its  place. 

2.  Can  a   body  put  in  motion  stop  itself? 

i^It  has  been  observed  in  Art.  11,  that/ bodies  are  indiffer- 
ent to  rest  or  motion.  Hence,  a  body  cannot  put  itself  in 
motion,  or  stop  itself  after  it  has  begun  to  move. 

3.  What  is  force  or  power? 

That  which  puts  a  body  in  motion,  or  which  changes 
its  motion  after  it  has  begun  to  move,  is  called  force  or 
power.  ""Thus,  the  stroke  of  the  hammer  is  the  force  which 
drives  the  nail,  the  effort  of  the  horse  the  force  which 
moves  the  carriage,  and  the  attraction  of  gravitation  the 
force  which  draws  bodies  to  the  earth. 

4.  What  is  velocity/  ? 

The  rate  at  which  a  body  moves,  or  the  rapidity  of  its. 
motion,  is  estimated  by  the  space  which  it  passes,  over  in 


220  BOOK    VII. SECTION    II. 

a  given  portion  of  time,  and  this  rate  is  called  its  velo- 
city. Thus,  if  in  one  minute  of  time  a  body  passes  over 
200  feet,  its  velocity  is  said  to  be  200  feet  per  minute  ; 
and  if  another  body,  in  the  same  time,  passes  over  400 
feet,  its  velocity  is  said  to  be  400  feet  per  mirmte,  or 
double  that  of  the  first. 

5.  What  is  uniform  velocity  ?  ' 

When  a  body  moves  over  fequal  distances  in  equal  times, 
its  velocity  is  said  to  be  uniform.  Thus,  if  a  body  move 
at  the  rate  of  30  feet  a  second,  it  has  a  uniform  velocity, 
for  it  always  passesover  an  equal  space  in  an  equal  time. 

6.  What  is  uniformly-accelerated  velocity  ? 

Bodies  which  receive  uniform  accelerations  of  velocity, 
that  is,  equal  accelerations  in  equal  times,  are  said  to  have 
motions  uniformly  accelerated. 

7.  How  will  a  body  fall  by  the  attraction  of  gravitation  ? 
If  a  body  fall  freely  towards  the  earth,  by  the  attractioji 

of  graAritation,  it  will  descend  in  a  line  perpendicular  to  its 
surface. 

8.  How  far  will  it  fall  in  the  frst  second,  and  how  far 
in  each  succeeding  second?  What  kind  of  velocity  will  it 
Juive? 

In  the  first  second  it  will  fall  through  16  feet;  in  the 
second  second,  having  the  velocity  already  acquired,  and 
being  still  acted  on  by  the  force  of  gravity,  it  will  descend 
through  32  feet ;  in  the  third  second  it  will  descend  through 
48  feet ;  in  the  fourth  second  through  64  feet,  and  so  on, 
adding  to  its  velocity  in  every  additional  second.  This  is 
a  motion  uniformly  accelerated,  for  the  velocity  is  equally 
increased  in  each  second  of  time. 


LAWS    OF    MOTION,    ETC.  221 

9.  What  is  momentum? 

Momentum  is  the  force  with  which  a  body  in  motion 
would  strike  against  another  body.  If  a  body  of  a  given 
weight,  say  10  pounds,  were  moving  at  the  rate  of  30  feet 
per  second,  and  another  body  of  the  same  weight  were  to 
move  twice  as  fast,  the  last  would  have  double  the  mo- 
mentum of  the  first. 

10.  On  what  does  the  momentum  of  a  body  depend? 
When  the  bodi'es  are  of  a  given  weight,  the  momentum 

will  depend  on  the  velocity.  But  if  two  unequal  bodies 
move  with  the  same  velocity,  their  momentum  will  depend 
upon  their  weight.  Hence,  the  momentum  of  a  body  will 
depend  on  its  weight  and  velocity ;  that  is,  it  will  be  equal 
to  the  weight  multiplied  by  the  velocity. 

If  the  weight  of  a  body  be  represented  by  5,  and  its  ve- 
locity by  6,  its  momentum  will  be  5  x  6  =  30. 

If  the  weight  of  a  body  be   represented  by  8,  and  its 
velocity  by  2,  its  momentum  will  be  represented  .by 
16  X  2  =  32. 

11.  What  are  action  and  reaction,  and  how  do  they  com- 
pare with  each  other  ? 

When  a  body  in  motion  strikes  against  another  body, 
it  meets  with  resistance.  The  force  of  the  moving  body 
is  called  action,  and  the  resistance  offered  by  the  body 
struck  is  called  reaction ;  and  it  is  a  general  principle,  that 
action  and  reaction  are  equal.  Thus,  if  you  strike  a  nail 
with  a  hammer,  the  action  of  the  hammer  against  the  nail 
is  just  equal  to  the  reaction  of  the  nail  against  the  ham- 
mer. Also,  if  a  body  fall  to  the  earth,  by  the  attraction 
of  gravitation,  the  action  of  the  body  when  it  strikes  the 
earth  is  just  equal  to  the  reaction  of  the  earth  against  the 
body. 


222  BOOK  vri. — section  ii. 

12.   What  is  the  centre  of  gravity  ? 

The  centre  of  gravity  is  that  point  of  a  body  about  which 
all  the  parts  will  exactly  balance  each  other.  Hence,  if 
the  centre  of  gravity  be  supported,  the  body  will  not  fall, 
for  all  the  parts  will  balance  each  other  about  the  centre 
of-  gravity. 


13.  Is  the  centre  of  gravity  changed  hy  changing  the  po- 
sition of  a  body  ? 

'The  centre  of  gravity  of  a  body  is  not  changed  by 
changing  the  position  of  the  body.  Thus,  if  a  body  be 
suspended  by  a  cord,  attached  at  its  centre  of  gravity,  it 
will  remain  balanced,  in  every  position  of  the  body. 


14.  -/jT  two  equal  bodies  are  joined  by  a  bar,  where  will  be 
the  centre  of  gravity  ? 

If  we  have  two  equal  bodies 
A  and  B,  connected  together  by 
a  bar  AB,  the  centre  of  gravity 
will  be  at  C,  the  middle  point  of  AB,  and  about  this  point 
the  bodies  will  exactly  balance  each  other. 


15.  If  the  bodies  are  unequal,  where  will  it  be  found? 

If  we  have  two  unequal 
bodies  A  and  B,  the  cen- 
tre of  gravity  C  will  be 
nearer  the  larger  body  than 
the  smaller,  and  just  as 
much  nearer  as  the  larger 
body  exceeds  the  smaller.  Thus,  if  B  is  three  times  greater 
than  A,  then  BC  will  be  one-third  of  AC. 


CENTRE    OF    GRAVITY. 


223 


16.  If  one  body  is  very  large  in  comparison  with  the  other t 
what  will  take  place? 

If  one  of  the 
bodies  is  very  large 
in  comparison  with 
the  other,  the  cen- 
tre of  gravity  may 
fall  within  the  larger 

body.     Thus,  the  centre  of  "gravity  of  the  bodies  A  and  B 
falls  at  C.  '     ' 

17.  What  is  the  line  of  direction  of  the  centre  of  gravity  ? 
The  vertical  line   drawn  through  the   centre  of  gravity, 

is  called  the  line  of  direction  of  the  centre  of  gravity. 

18.  If  this  line  is  supported,  will  the  body  fall  ? 
If  the  line  of  direction  of  the  centre 

of  gravity  falls  within'  the  base  on 
which  the  body  stands,  the  body  will 
be  supported ;  but  if  the  line  falls  with- 
out the  base,  the  body  will  fall.  Thus, 
if  in  a  wine  glass,  the  centre  of  gravity 

be  at  C,  the  glass  will  fall  the  moment  the  line  CD  falls 
without  the  base. 

19.  If  the  line  of  direction  of  the   centre  of  gravity  falls 
near  the  base,  is  the  body  likely  to  fall  1     Give  the  illustration. 

Let  us  suppose  a  cart  on  in- 
clined ground  to  be  loaded  with 
stone,  so  that  the  centre  of  gravity 
of  the  mass  shall  fall  at  C.  In 
this  position  the  line  of  direction 
CD  falls  within  the  base,  and  the 
cart  will  stand.  But  if  the  cart 
be  loaded  with  hay,  so  as  to  bring 


B  n 


224  BOOK    VII. SECTION    III. 

the   centre  of  gravity  at  A,  the  line  of  direction  AB  will 
fall  without  the  base,  and  the  cart  will  be  upset. 


SECTION   III. 

OF    THE    MECHANICAL    POWERS. 


1,  How  many  mechanical  powers  are  there,  and   what  are 
? 


There  are  six  simple  machines,  which  are  called  Mechan- 
ical powers.  They  are  the  Lever,  the  Pulley,  the  Wheel  and 
Axle,  the  Inclined  Plane,  the  Wedge,  and  the  Screw. 

2.  What  four  things  must  be  considered,  m  order  to  un- 
derstand the  power  of  a  machine  ? 

1st.  The  power  or  force  which  acts.  This  consists  in 
the  effort  of  men  or  horses,  of  weights,  springs,  steam,  &c. 

2d.  The  resistance  which  is  to  be  overcome  by  the 
power.     This  generally  is  a  weight  to  be  moved. 

3d.  We  are  to  consider  the  centre  of  motion,  or  fil- 
crum,  which  means  a  prop.  The  prop  or  fulcrum  is  the 
point  about  which  all  the  parts  of  the  machine  move. 

4th.  We  are  to  consider  the  respective  velocities  of  the 
power  and  resistance. 

3.  When  is  a  machine  said  to  he  in  equilibrium? 

A  machine  is  said  to  be  in  equilibrium  when  the  resist- 
ance exactly  balances  the  power,  in  which  case  all  the 
parts  of  the  machine  are  at  rest. 

4.  What  is  a  Lever? 

The  lever  is  a  straight  bar  of  wood  or  metal,  which 
moves  around  a  fixed  point  called  the  fulcrum. 


OF    THE    MECHANICAL    POWERS. 


225 


5.  Hfiw  many  kinds  of  levers  are  there  ? 
-There  are  three  kinds  of  levers: 
1st.   When  the  fulcrum  is  i  ~~ 

between  the  weight  and  the        f<v>\ 
power.  "^ 


1 


2d.  "When  the  weight  is 
between  the  power  and  the 
fulcrum. 


3d.  When  the  power  is 
between  the  fulcrum  and  the 
weight. 


"6.    What  are  the  arms  of  a  lever? 

The  parts  of  the  lever,  from  the  fulcrum  to  the  weight 
and  power,  are  called  the  arms  of  the  lever. 

7,  -  When  is  an  equilibrium  produced  in  the  lever  ? 

An  equilibrium  is  produced  in  all  the  levers  when  the 
weight,  multiplied  by  its  distance  from  the  fulcrum,  is  equal 
to  the  product  of  the  power  multiplied  by  its  distance  from 
the  fulcrum.     That  is, 

The  weight  is  to  the  power ^  as  the  distance  from  the  power 
to  the  fulcrum,  to  distance  from  the  weight  to  the  fulcrum. 


EXAMPLES. 


1.  In  a  lever  of  the  first  kind  the  fulcrum  is  placed  at 
the  middle  point:  what  power  will  be  necessary  to  bal- 
ance a  weight  of  40  pounds  ?  ,     - 

10* 


226  BOOK    VII. SECTION    III. 

2.  In  a  lever  of  the  second  kind,  the  weight  is  placed 
at  the  middle  point :  what  power  will  be  necessary  to  sus- 
tain a  weight  of  50  lbs.  ? 

3.  In  a  lever  of  the  third  kind,  the  power  is  placed  at 
the  middle  point :  what  power  will  be  necessary  to  sus- 
tain a  weight  of  25  lbs.  ? 

4.  A  lever  of  the  first  kind  is  8  feet  long,  and  a  weight 
of  60  lbs.  is  at  a  distance  of  2  feet  from  the  fulcrum : 
what  power  will  be  necessary  to  balance  it  ? 

Ans.  20  lbs, 

5.  In  a  lever  of  the  first  kind,  that  is  6  feet  long,  a 
weight  of  200  lbs.  is  placed  at  1  foot  from  the  fulcrum: 
what  power  will  balance  it  ? 

Ans.  40  lbs. 

6.  In  a  lever  of  the  first  kind,  like  the  common  steel- 
yard, the  distance  from  the  weight  to  the  fulcrum  is  one 
inch :  at  what  distance  from  the  fulcrum  must  the  poise 
of  1  lb.  be  placed,  to  balance  a  weight  of  1  lb.  ?  A  weight 
of  11  lbs.  ?     Of  2  lbs.  ?     Of  4  lbs.  ? 

7.  In  a  lever  of  the  third  kind,  the  distance  from  the 
fulcrum  to  the  power  is  5  feet,  and  from  the  fulcrum  to 
the  weight  8  feet:  what  power  is  necessary  to  sustain  a 
weight  of  40  lbs.  ? 

Ans.  64  lbs. 

8.  In  a  lever  of  the  third  kind,  the  distance  from  the 
fulcrum  to  the  weight  is  12  feet,  and  to  the  power  8  feet: 
what  power  will  be  necessary  to  sustain  a  weight  of  100 
lbs.? 

Ans.  150  lbs. 

8.  How  are  the  equilibriums  of  levers  affected  by  consider- 
ing their  weight  ? 

In  levers  of  the  first  kind,  the  weight  of  the  lever  gener- 


OF    THE    PULLEY 


227 


ally  adds  to  the  power,  but  in  the  second  and  third  kinds, 
the  weight  goes  to  diminish  the  effect  of  the  power. 

9.  What  has  been  stated  in  the  previous  examples  ?  What 
is  necessary  that  the  machine  may  move  ? 

In  the  previous,  examples,  we  have  stated  the  circum- 
stances under  which  the  power  will  exactly  sustain  the 
weight.'  In  order  that  the  power  may  overcome  the  re- 
sistance, it  must  of  course  be  somewhat  increased.  The 
lever  is  a  very  important  mechanical  power,  being  much 
used,  and  entering  indeed  into  all  the  other  machines. 


OF    THE    PULLEY. 


10,    what  is  a  pulley  ? 

The  pulley  is  a  wheel,  having  a  groove 
cut  in  its  circumference,  for  the  purpose 
of  receiving  a  cor^i  which  passes  over  it. 
When  motion  is  imparted  to  the  cord,  the 
pulley  turns  around  its  aiis,  which  is 
generally  supported  by  being  attached  to 
a  beam  above. 


r.\ 


m    9 


11,  How  many  kinds  of  pulleys  are  there? 

Pulleys  are   divided  into  two  kinds,  fixed  pulleys  and 
moveable  pulleys. 


12.  Does  a  fixed  pulley  increase  the  power? 

When  the  pulley  is  fixed,  it  does  not  increase  the  power 
which  is  applied  to  raise  the  weight,  but  merely  changes 
the  direction  in  which  it  a«t8. 


228 


BOOK    VII. SECTION    III. 


13.  Does  a  moveable  pulley  give  any  ad- 
vantage in  power? 

A  moveable  pulley  gives  a  meclianical 
advantage.  Thus,  in  the  moveable  pulley, 
the  hand  which  sustains  the  cask  does  not 
actually  support  but  one  half  the  weight 
of  it;  the  other  half  is  supported  by  the 
hook  to  which  the  other  end  of  the  cord 
is  attached. 


14.  Will  an  advantage  be  gained  by  several 
moveable  pulleys  ?      What  will  be  lost  1 

If  we  have  several  moveable  pulleys  the 
advantage  gained  is  still  greater,  and  a  very 
heavy  weight  may  be  raised  by  a  small  power. 
A  longer  time,  however,  will  be  required,  than 
with  a  single  pulley.  It  is  indeed  a  general 
principle  in  machines,  that  what  is  gained  in 
power  is  lost  in  time,  and  this  is  true  for  all 
machines. 


^ 


15.  Is  there  an  actual  loss  of  power?  What  does  it  arise 
from  ? 

There  is  also  an  actual  loss  of  power,  viz.,  the  resist- 
ance of  the  machine  to  motion,  arising  from  the  rubbing 
of  the  parts  against  each  other,  which  is  called  the  fric- 
tion of  the  machine.  This  varies  in  the  different  machines, 
but  must  always  be  allowed  ibr,  in  calculating  the  powei 


OF    THE    PULLEY.  229 

necessary  to  do  a  given  work.  It  would  be  wrong,  how- 
ever, to  suppose  that  the  loss  was  equivalent  to  the  gain, 
and  that  no  advantage  is  derived  from,  the  mechanical 
powers.  We  are  unable  to  augment  our  strength,  but  by 
the  aid  of  science  we  so  divide  the  resistance,  that  by  a 
continued  exertion  of  power  we  accomplish  that  which  it 
would  be  impossible  to  effect  by  a  single  effort. 

If  in  attaining  this  result  we  sacrifice  time,  we  cannot 
but  see  that  it  is  most  advantageously  exchanged  for  power. 


16.  In  the  moveable  pulley,  what  proportion  exists  between 
the  power  and  the  weight? 

It  is  plain  that,  in  the  moveable  pulley,  all  the  parts  of 
the  cord  will  be  equally  stretched,  and  hence,  each  cord 
running  from  pulley  to  pulley  will  bear  an  equal  part  of 
the  weight ;  consequently,  the  power  will  always  be  equal  to 
the  weight,  divided  by  the  number  of  cords  which  reach  from 
•pulley  to  pulley. 

EXAMPLES. 

1.  In  a  single  immoveable  pulley,  what  power  will  sup- 
port a  weight  of  60  lbs.  ? 

2.  In  a  single  moveable  pulley,  what  power  will  support 
a  weight  of  80  lbs.  ? 

3.  In  two  moveable  pulleys  with  5  cords,  (see  last  fig.,) 
what  power  will  support  a  weight  of  100  lbs.  ? 

Ans.  20  lbs. 


230  BOOK    Vll. — SECTION    III. 


WHEEL    AND    AXLE. 


17.  Of  what  is  the  wheel  and  axle  composed?  How  is 
the  axle  supported? 

,  This  machine  is  com- 
posed of  a  wheel  or  crank, 
ifirmly  attached  to  a  cylin- 
drical axle.  The  axle  is 
supported -at  its  ends  by 
two  pivots,  which  are  of 
less  diameter  than  the  axle 
around  which  the  rope  is 
coiled,  and  which  turn 
freely  about  the  points  of 
support. 

18.  What  is  the  proportion  between  the  power  and  weight  ? 
'  In  o"rder  to  balance  the  weight,  we  must  have. 

The  power  to  the  weight,  as  the  radius  of  the  axle  to  the 
length  of  the  crank,  or  radius  of  the  wheel. 

EXAMPLES. 

1.  What  must  be  the  length  of  a  crank  or  radius  of  a 
wheel,  in  order  that  a  power  of  40  lbs.  may  balance  a 
weight  of  600  lbs.,   suspended  from  an  axle   of  6  inches 

'^^'"^'  •       ,  Ans.7ifl. 

2.  What  must  be  the  diameter  of  an  axle,  that  a  power 
of  100  lbs.  applied  at  the  circumference  of  a  wheel  of  6 
feet  diameter  may  balance  400  lbs.? 

Ans.  l^ft. 


INCLINED    PLANE.  231 


INCLINED    PLANE. 


19.  What  is  an  inclined  plane? 

The  inclined  plane  is  nothing  more  than  a  slope  or  de- 
clivity, which  is  used  for  the  purpose  of  raising  weights. 
It  is  not  difficult  to  see  that  a  weight  can  be  forced  up 
an  inclined  plane  more  easily  than  it  can  be  raised  in  a 
vertical  line.  But  in  this,  as  in  the  other  machines,  the 
advantage  is  obtained  by  a  partial  loss  of  power. 

20.  What  proportion  exists  between  the  power  and  the 
weight,  when  they  are  in  equilibrium  1 

If  a  weight  W  b^ 
supported  on  the  in- 
clined plane  ABC  by 
a  cord  passing  over  a 
pulley  at  F,  and  the 
cord  from  the  pulley  to 

the  weight  be  parallel  to  the  length  of  the  plane  AB^  the 
power  P  will  balance  the  weight  W,  when 

F  \  W  :   :  height  BC  :  length  AB. 

It  is  evident  that  the  power  ought  to  be  less  than  the 
weight,  since  a  part  of  the  weight  is  supported  by  the 
plane. 

EXAMPLES. 

1.  The  length  of  a  plane  is  30  feet,  and  its  height  6 
feet:  what  power  will  be  necessary  to  balance  a  weight 

of  200  lbs.?  A        Ann 

Ans.  40  lbs. 

2.  The  height  of  a  plane  is  10  feet,  and  the  lengtli  20 
feet:   what  weight  will  a  power  of  50  lbs.  support? 

Ans.  100  lbs. 


232 


BOOK    VII. SECTION    III. 


3.  The  height  of  a  plane  is  15  feet,  and  length  45  feet 
what  power  will  sustain  a  weight  of  180  lbs.  ? 

Ans.  60  lbs. 

THE    WEDGE. 

21.  What  is  the  wedge ^^  and  what  is  it  used  for 

The  wedge  is  composed  of  two  in- 
clined planes,  united  together  along 
their  bases,  and  forming  a  solid  ACB. 
It  is  used  to  cleave  masses  of  wood 
or  stone.  The  resistance  which  it 
overcomes  is  the  attraction  of  cohe- 
sion of  the  body  which  it  is  employed, 
to-  separate.  The  wedge  acts  principally  by  being  struck 
with  a  hammer  or  mallet  on  its  head,  and  very  little  effect 
can  be  produced  with  it,  by  mere  pressure. 

All  cutting  instruments  are  constructed  on  the  principle 
of  the  inclined  plane  or  wedge.  Such  as  have  but  one 
sloping  edge,  like  the  chisel,  may  be  referred  to  the  in- 
clined plane ;  and  such  as  have  two,  like  the  axe  and  the 
knife,  to  that  of  the  wedge. 

THE    SCREW. 

22.  Of  how  many  parts  is  the  screw  composed  ?  Describe 
its  parts  and  uses. 

The  screw  is  composed  of 
two  parts,  the  screw  S,  and 
the  nut  N. 

The  screw  aS  is  a  cylinder 
with  a  spiral  projection  wind- 
ing around  it,  called  the  thread. 
The  nut  N  is  perforated  to  ad- 
mit the  screw,  and  within  it  is 
a  groove  into  which  the  thread 
of  the  screw  fits  closely. 


THE    SCREW.  233 

The  handle  D,  which  projects  from  the  nut,  is  a  lever 
which  works  the  nut  upon  the  screw.  The  power  of  the 
screw  depends  on  the  distance  between  the  threads.  The 
closer  the  threads  of  the  screw,  the  greater  will  be  the 
power,,  but  then  the  number  of  revolutions  made  by  the 
handle  D  will  also  be  propohionally  increased;  so  that  we 
return  to  the  general  principle — what  is  gained  in  power 
is  lost  in  time.  The  power  of  the  screw  may  also  be  in- 
creased by  lengthening  the  lever  attached  to  the  nut. 

The  screw  is  used  for  compression,  and  to  raise  heavy 
weights.  It  is  used  in  cider  and  wine  presses,  in  coining, 
and  for  a  variety  of  other  purposes. 

aENERAL    REMARKS. 

All  machines  are  composed  of  one  or  more  of  the  six 
machines  which  we  have  described.  We  should  remem- 
ber, that  friction  diminishes  very  considerably  the  power 
of  machines. 

There  are  no  surfaces  in  nature  which  are  perfectly 
smooth.  Polished  metals,  although  they  appear  smooth,  are 
yet  far  from  being  so.  If,  therefore,  the  surfaces  of  two 
bodies  come  into  coutact,  the  projections  of  the  one  will 
fall  into  the  hollow  parts  of  the  other,  and  occasion  more 
or  less  resistance  to  motion.  In  proportion  as  the  surfaces 
of  bodies  are  polished,  the  friction  is  diminished,  but  it  is 
always  very  considerable,  and  it  is  computed  that  it  gen- 
erally destroys  one-third  the  power  of  the  machine. 

Oil  or  grease  is  generally  used  to  lessen  the  friction. 
It  fills  up  the  cavities  of  the  rubbing  surfaces,  and  thus 
makes  them  slide  more  easily  over  each  other. 


234  BOOK    VII. — SECTION    IV. 


SECTION  IV. 

OF    SPECIFIC    GRAVITY. 

1.  What  is  the  specific  gravity  of  a  body  ? 

The  specific  gravity  of  a  body  is  the  relation  which  the 
weight  of  a  given  magnitude  of  that  body  bears  to  the 
weight  of  an  equal  magnitude  of  a  body  of  another  kind. 

2.  When  is  one  body  said  to  be  specifically  heavier  than 
another  ? 

If  two  bodies  are  of  the  same  bulk,  the  one  which  weighs 
the  most  is  said  to  be  specifically  heavier  than  the  other. 
On  the  contrary,  one  body  is  said  to  be  specifically  lighter 
than  another,  when  a  certain  bulk  or  volume  of  it  weighs 
less  than  an  equal  bulk  of  that  other. 

Thus,  if  we  have  two  equal  spheres,  each  one  fool  in 
diameter,  the  one  of  lead  and  the  other  of  wood,  the  leaden 
one  will  be  found  to  be  heavier  than  the  wooden  one  ;  and 
hence,  its  specific  gravity  is  greater.  On  the  contrary,  the 
wooden  sphere  being  lighter  than  the  leaden  one,  its  specific 
gravity  is  less.  % 

3.  What  does  the  greater  specific  gravity  indicate  1  What 
is  density? 

The  greater  specific  gravity  of  a  body  indicates  a  greater 
quantity  of  matter  in  a  given  bulk,  and  consequently  the 
matter  must  be  more  compact,  or  the  particles  nearer  to- 
gether. This  closeness  of  the  particles  is  called  density. 
Hence,  if  two  bodies  are  of  equal  bulk  or  volume,  their 
weights  or  specific  gravities  will  be  proportional  to  their 
densities. 


OP    SPECIFIC    GRAVITY.  235 

4.  If  two  bodies  are  of  the  same  specific  gravity,  how  will 
the  weights  be  ? 

If  two  bodies  are  of  the  same  specific  gravity,  or  density, 
their  weights  will  be  proportional  to  their  bulks. 

5.  If  a  body  be  i?nmersed  in  a  fluid,  what  will  take  place  ? 
If  the  body  is  specifically  heavier  than  the  fluid,  it  will 

sink  on  being  immersed.  It  will,  however,  descend  less 
rapidly  through  the  fluid  than  through  the  air,  and  less 
power  will  be  required  to  sustain  the  body  in  the  fluid  than 
out  of  it.  Indeed,  it  will  lose  as  much  of  its  weight  as  is 
equal  to  the  weight  of  a  quantity  of  fluid  of  the  same  bulk. 
If  the  body  is  of  the  same  specific  gravity  with  the  fluid, 
it  loses  all  its  weight,  and  requires  no  force  but  the  fluid 
to  sustain  it.  If  it  be  lighter,  it  will  be  but  partially  im- 
mersed, and  a  part  of  the  body  will  remain  above  the  sur- 
face of  the  fluid. 

6.  What  do  we  conclude  from  the  preceding  article  ? 

1st.  That  when  a  heavy  body  is  weighed  in  a  fluid,  its 
weight  will  express  the  difference  between  its  true  weight 
and  that  of  an  equal  bulk  of  the  fluid. 

2d.  If  the  body  have  the  same  specific  gravity  with  the 
fluid,  its  weight  will  be  nothing. 

3d.  If  the  body  be  lighter  than  the  fluid,  it  will  require 
a  force  equal  to  the  difference  between  its  own  weight 
and  that  of  an  equal  bulk  of  the  fluid  to  keep  it  entirely 
immersed,  that  is,  to  overcome  its  tendency  to  rise, 

7.  What  is  necessary  in  comparing  the  weights  of  bodies  ? 
In  comparing  the  weights  of  bodies,  it  is  necessary  to 

take  some  one  as  a  standard,  with  which  to  compare  all 
others. 


236  BOOK    VII. SECTION    IV. 

8.  What  is  generally  taken  as  the  standard? 
Rain-water  is  generally  taken  as  this  standard. 

9.  What  is  the  weight  of  a  cubic  foot  of  rain-water  ? 

A  cubic  foot  of  rain-water  is  found,  by  repeated  experi- 
ments, to  weigh  62i  pounds,  avoirdupois,  or  1000  ounces. 
Now,  since  a  cubic  foot  contains  1728  cubic  inches,  it  fol*- 
lows  that. one  cubic  inch  weighs  .03616898148  of  a  pound. 
Therefore,  if  the  specific  gravity  of  any  body  be  multiplied 
by  .03616898148,  the  product  will  be  the  weight  of  a  cubic 
inch  of  that  body  in  pounds  avoirdupois.  And  if  this  weight 
be  then  multiplied  by  175,  and  the  product  divided  by  144, 
the  quotient  will  be  the  weight  of  a  cubic  inch  in  pounds 
troy;  since  144  lbs.  avoirdupois  is  just  equal  to  175  lbs. 
troy. 

10.  How  will  the  specific  gravity  of  a  body  be  to  that  of 
the  fluid  in  which  it  is  immersed  ? 

Since  the  specific  gravities  of  bodies  are  as  the  weights 
of  equal  bulks,  the  specific  gravity  of  a  body  will  be  to 
th^  specific  gravity  of  a  fluid  in  which  it  is  immersed,  as 
the  true  weight  of  the  body  to  the  weight  lost  in  weighing 
it  in  the  fluid.  Hence,  the  specific  gravities  of  different  fluids 
are  to  each  other  as  the  weights  lost  by  the  same  solid  im- 
mersed in  them. 

11.  How  do  you  flnd  the  specific  gravity  of  a  body,  when 
the  body  is  heavier  than  water? 

1st.  Weigh  the  body  first  in  air  and  then  in  rain-water, 
and  take  the  diflference  of  the  weights,  which  is  the  weight 
lost. 

2d.  Then  say,  as  the  weight  lost  is  to  the  true  weight, 
so  is  the  specific  gravity  of  the  water  to  the  specific  grav- 
ity of  the  body. 


OF    SPECIFIC    GRAVITY.  237 


EXAMPLES. 


1.  A  piece  of  platina  weighs  70.5588  lbs.  in  the  air, 
and  in  water  only  66.9404  lbs. :  what  is  its  specific  gravity, 
that  of  water  being  taken  at  1000? 

First,  70.5588  —  66.9404  =  3.6184  lost  in  water. 

Then,    3.6184  :  70.5588  :  :   1000  :   19500,   which   is    the 

specific  gravity,  or  weight  of  a  cubic  foot  of  platina. 

2,  A  piece  of  stone  weighs  10  lbs.  in  air,  but  in  water 
only  6^  lbs.  :   what  is  its  specific  gravity  ? 

Ans.   3077. 

12.  How  do  you  Jlnd  the  specific  gravity  of  a  body  when 
it  is  lighter  than  water? 

1st.  Attach  another  body  to  it  of  such  specific  gravity, 
that  both  may  sink  in  the  water  as  a  compound  mass. 

2d.  Weigh  the  heavier  body  and  the  compound  mass 
separately,  both  in  water  and  in  open  air,  and  find  how 
much  each  loses  by  being  weighed  in  water. 

3d.  Then  say,  as  the  difference  of  these  losses  is  to  the 
weight  of  the  lighter  body  in  the  air,  so  is  the  specific 
gravity  of  water  to  the  specific  gravity  of  the  lighter  body. 

EXAMPLES. 

1.  A  piece  of  elm  weighs  15  lbs.  in  open  air.  A  piece 
of  copper  which  weighs  18  lbs.  in  air  and  16  in  water  is 
attached  to  it,  and  the  compound  weighs  6  lbs.  in  water: 
what  is  the  specific  gravity  of  the  elm? 

Copper.  Compound. 

18  in  air.  33  in  air. 

16  in  water.  6  in  water. 

2  loss.  27  loss. 

Then,  27  — 2  =25  =  difl!erence  of  losses. 


238  BOOK    VII.— SECTION    IV. 

Then,  as  25  :  15  :  :  1000  :  600,  which  is  the  specific 
gravity  of  the  elm. 

2.  A  piece  of  cork  weighs  20  lbs.  in  air,  and  a  piece  of 

granite  weighs  120  lbs.  in  air,  and  80  lbs.  in  water.     When 

the    granite   is   attached  to  the   cork   the   compound  mass 

weighs  16f  lbs.  in  water:   what  is  the  specific  gravity  of 

the  cork?  a        nAn 

Ans.  240. 

13.  How  do  you  find  the  specific  gravity  of  fiuids? 

1st.  Weigh  any  body  whose  specific  gravity  is  known, 
both  in  the  open  air,  and  in  the  fluid,  and  take  the  difier- 
ence,  which  is  the  loss  of  weight. 

2d.  Then  say,  as  the  true  weight  is  to  the  loss  of  weight, 
so  is  the  specific  gravity  of  the  solid  to  the  specific  grav- 
ity of  the  fluid. 

EXAMPLES. 

1.  A  piece  of  iron  weighs  298.1  ounces  in  the  air,  and 
259.1  ounces  in  a  fluid;  the  specific  gravity  of  the  iron  is 
7645  :    what  is  the  specific  gravity  of  the  fluid  ? 

First,  298.1  —  259.1  =r  39  loss  of  weight: 

Then,  298.1  :  39  ::  7645  :  1000,  which  is  the  specific  grav- 
ity of  the  fluid :  hence  the  fluid  is  water. 

2.  A  piece  of  lignumvitge  weighs  42|  ounces  in  a  fluid, 
and  166|  ounces  out  of  it :  what  is  the  specific  gravity  of 
the  fluid — that  of  lignumvitae  being  1333  ? 

Ans.  991,  which  shows  the  fluid  to  be  liquid  turpentine 
or  Burgundy  wine. 

Note. — In  a  similar  manner  the  specific  gravities  of  all 
liquids  may  be  found  from  the  following  table. 


OP    SPECIFIC    GRAVITIES. 


239 


TABLE    OF    SPECIFIC    GRAVITIES. 


Sp.gr.  wt.  cub.in. 

Sp.gr.  wt.  cub.fl. 

Platina,  hammered  20.336  - 

11.777 

Ebony 

1.331  - 

83.18 

Platina 

19.500  - 

11.285 

Oak,  60  years  old 

1.170  - 

73.12 

Pure  cast  gold 

19.258  - 

11.145 

Amber 

1.078 

Mercury- 

13.568  - 

7.872 

Beer 

1.034 

Cast  lead 

11.352  - 

6.569 

Milk 

1.030 

Pure  cast  silver 

10.474  - 

6.061 

Sea  water 

1.028 

Cast  copper 

8.788  - 

5.085 

Distilled  water 

1.000 

Cast  brass 

8.395  - 

4.856 

Liquid  turpentine 

.991 

Hard  steel 

7.816  - 

4.523 

Burgundy  wine 

.991 

Cast  cobalt 

7.811  - 

4.520 

Camphor 

.989 

Cast  nickel 

7.807  - 

4.513 

Oak,  English^ 

.970  - 

60.62 

Bar  iron 

7.788  - 

4.507 

Bees'  wax 

.965 

Cast  tin 

7.291  - 

4.219 

Tallow 

.945 

Cast  iron 

7.207  - 

4.165 

Olive  oil 

.915 

Cast  zinc 

7.190  - 

4.161 

Logwood 

.913  - 

57.06 

wt.  cub.  ft. 

Box,  French 

.912  - 

57.00 

lbs. 

Wax 

.897 

Limestone 

3.179  - 

198.68 

Oak,  Canadian 

.872  - 

54.50 

White  glass 

2.892 

Alder 

.800  - 

50.00 

Chalk 

2.784  - 

174.00 

Apple  tree 

.793  - 

49.56 

Marble 

2.742  - 

171.38 

Ash  and  Dantzic  oak    .760  - 

47.50 

Alabaster 

2.730 

Maple  and  Riga  fir 

.750  - 

46.87 

Pearl 

2.684 

Cherry  tree 

.715  - 

44.68 

Slate 

2.672  - 

167.00 

Beech 

.696  - 

43.50 

Pebble 

2.664  - 

166.50 

Elder  tree 

.695  - 

43.44 

Green  glass 

2.642 

Walnut 

.671  - 

41.94 

Flint  and  spar 

2.594  - 

162.12 

Pear  tree 

.661  - 

41.31 

Common  stone 

2.520  - 

157.50 

Pitch  pine 

.660  - 

41.25 

Paving  stones 

2.416  - 

151.00 

Cedar 

.596  - 

37.25 

Sulphur 

2.033  - 

127.06 

Mahogany 

.560  - 

35.00 

Brick 

2.000  - 

125.00 

Elm  and  West  India  fir  .556  - 

34.75 

Ivory 

1.822 

Larch 

.544  - 

34.00 

Bone  of  an  ox 

1.659 

Poplar 

.383  - 

23.94 

Honey 

1.456 

Cork 

.240  - 

15.00 

Lignum  vitcB 

1.333  - 

83.31 

Air  at  the  earth's  surf.    .00 if 

Remark. — In  the  table  of  specific  gravities,  the  cubic 
foot  of  water,  which  weighs  1000  ounces,  is  taken  as  the 
standard,  and  the  figures  in  the  column  of  specific  gravity 
show  how  many  times  each  substance  is  heavier  or  lighter 
than  water.  If  the  number  opposite  each  substance  be 
multiplied  by  1000,  the  product  will  be  the  weight  of  a 
cubic  foot  of  that  substance,  in  ounces.  The  other  column 
shows  the  weight  in  ounces  of  a  cubic  inch,  or  the  weight 
in  pounds  of  a  cubic  foot. 


240  BOOK    VII. SECTION    IV. 

14.  How  do  you  find  the  solidity  of  a  body  when  its  weight 
and  specific  gravity  are  given  ? 

As  the  tabular  specific  gravity  of  the  body  is  to  its 
weight  in  ounces  avoirdupois,  so  is  1  cubic  foot  to  the  con- 
tent in  cubic  feet. 

EXAMPLES. 

1.  What  is  the  solid  content  of  a  block  of  marble,  that 
vreighs  10  tons,  its  specific  gravity  being  2742? 

First,  10  tons  —  358400  ounces. 
Then,  2742  :  358400  ::  1  :  130,-9JyV>  wliich  is  the   content 
in  cubic  feet. 

Note. — If  the  answer  is  to  be  found  in  cubic  inches, 
multiply  the  ounces  by  1728. 

2.  How  many  cubic  inches  in  an  irregular  block  of 
marble,  which  weighs  112  pounds,  allowing  its  specific 
gravity  to  be  2520? 

3.  How  many  cubic  inches  of  gunpowder  are  there  in 
1  pound  weight,  its  specific  gravity  being  1745? 

Ans.  15f,  nearly. 

4.  How  many  cubic  feet  are  there  in  a  ton  weight  of 
dry  oak,  its  specific  gravity  being  925? 

Ans.  38|f|. 


THE    END. 


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